m: 


ll-'.:v::''>.: 


(2,  I  3  [  ^fiole  Number  107] 

BUREAU   OF   EDUCATION 
CIRCULAR  OF  INFORMATION  NO.   3,  1890 


THE 


TEACHING  AND   HISTORY 


OF 


"^'^*^natt<tiMwg 


IN 


THE  UNITED   STATES 


BY 


-^ 


FLORIAN  CAJORI,  M.  S.  (University  of  Wisconsin) 

Formerly  Professor  op  Applied  Mathematics  in  the  Tulank  University  of 
Louisiana,  now  Professor  of  Physics  in  Colorado  College. 

r?*'H';x.>..   ru:  LEGE  Lli^iiAliy 


< 


-     -I'T  HILL,  MASS. 

MATH.  DEPT. 


WASHINGTON" 

GOVERNMENT'     PRINTING     OFFICB5 
1890 


.  Department  of  the  Interior, 

Bureau  of  Education, 
WasMngton,  D.  C,  February  19,  1889. 

Sir:  I  have  the  honor  to  transmit  herewith  the  manuscript  of  a  His- 
tory of  Mathematical  Teaching  in  the  United  States,  by  Prof.  Floriau 
Cajori,  a  graduate  of  the  University  of  Wisconsin,  student  at  Johns 
Hopkins  University,  and  recently  professor  of  applied  mathematics  in 
Tulane  University  of  Louisiana — a  work  prepared  with  your  approval, 
under  the  direction  of  this  Office. 

The  table  of  contents  indicates  the  wide  scope  of  the  work  and  the 
variety  of  subjects  treated,  but  scarcely  more  than  suggests  the  pains- 
taking labor  involved  in  its  preparation.  Professor  Cajori's  researches 
have  extended  through  several  years,  and  have  been  pursued  in  the 
libraries  of  Baltimore,  Philadelphia,  and  Washington.  He  has  person- 
ally conducted  a  large  correspondence  with  alumni,  and  past  and  pres- 
ent instructors  in  the  higher  educational  institutions,  and  has  been 
aided  by  1,000  circulars  of  inquiry  sent  from  this  Office  relating  to  the 
present  condition  of  mathematical  teaching  in  schools  of  all  grades. 

I  am  convinced  that  this  monograph  will  prove  of  great  value  to  all 
teachers  and  students  of  mathematics,  and  will  be  not  without  interest 
to  any  person  engaged  in  the  work  of  education.  I  therefore  respect- 
fully recommend  its  publication. 

I  have  the  honor  to  be,  sir,  very  respectfully,  your  obedient  servant, 

IS.  H.  E.  Dawson, 

Commissioner. 

Hon.  Wm.  F.  Yilas, 

Becretari/  of  the  Interior,  Washington,  D.  (7, 

I  3 


DEPA-RTlVrENT   OP  THE  INTERIOR, 

Washingt07i,  D.  C,  April  11,  1889. 
The  COMMTSSIONEB  OF  EDUCATION: 

Sir  :  1  acknowledge  the  receipt  of  your  letter  of  February  19,  1889, 
in  which  you  recommend  the  publication  of  a  monograph,  a  history  oi 
mathematical  teaching  in  the  United  States. 

Authority  is  hereby  given  for  the  publication  of  the  monograph,  pro- 
vided there  are  funds  in  sufficient  amount  available  for  such  purpose, 
Yery  respectfully, 

John  W..  Ij^oble, 

Secretary. 
5 


Digitized  by  the  Internet  Archive 

in  2010  with  funding  from 

Boston. Library  Consortium  IVIember  Libraries 


http://www.archive.org/details/teachinghistoryoOOcajo 


CONTENTS, 


Page. 

I.  Colonial  Times 9 

(a)  Elementary  Schools 9 

(b)  Colleges , 18 

Harvard  College 18 

Yale  College 28 

William  and  Mary  College 33 

University  of  Pennsylvania 36 

(c)  Self-tauglit  Mathematicians 37 

II.  Influx  op  English  Mathematics , 44 

(a)  Elementary  Schools 45 

(&)  Colleges - 55 

Harvard  College 57 

Yale  College 61 

University  of  Pennsylvania 65 

College  of  New  Jersey  (Princeton) 71 

Dartmouth  College 73 

Bowdoin  College 75 

Georgetown  College "h 

University  of  North  Carolina 77 

University  of  Sonth  Carolina i 81 

Kentucky  University 83 

United  States  Military  Academy 84 

(c)  Self-taught  Mathematicians 86 

(d)  Surveying  of  Government  Lands 92 

(e)  Mathematical  Journals 94 

III.  Influx  of  French  Mathematics 98 

(a)  Elementary  Schools 106 

{!))  Colleges — United  States  Military  Academy 114 

Harvard  College 127 

Yale  College 151 

College  of  New  Jersey 160 

Dartmouth  College 165 

Bowdoin  College „ 170 

Georgetown  College 173 

Cornell  University ,  176 

Virginia  Military  Institute 188 

University  of  Virginia ...„ 191 

University  of  North  Carolina 204 

University  of  South  Carolina „.., 208 

University  of  Alabama ..=  ..„. ., „,„ 214 

University  of  Mississippi.... .... .,  ..„.  ..„„..  „„„„  „.. 219 

Kentucky  Uoiversity ...„ .  ...o...... 225 

University  of  Tennessee ,..•...»*.....,....„....„ , 227 


8  CONTENTS. 

Page. 

III.  IiOFLUX  OF  Frejs^ch  MATHEMATICS— Colleges— Continued. 

Tulane  University  of  Louisiana 231 

University  of  Texas 236 

Washington  University 239 

University  of  Michigan 244 

University  of  Wisconsin 253 

Johns  Hopkins  University 261 

(c)  Mathematical  Journals 277 

(d)  U.  S.  Coast  and  Geodetic  Survey o 286 

IV.  The  Math  km  ATI  cal  Teaching  at  the  Present  Time ,...  293 

V.  Historical  Essays: 

(a)  History  of  Infinite  Series 361 

(ft)  On  Parallel  Lines  and  Allied  Subjects 376 

(c)  On  the  Foundation  of  Algebra 385 

(d)  Difference  between  Napier's  and  Natural  Logarithms 388 

(e)  Circle  Squarers - 391 

APPENDIX. 

Bibliography  of  Fluxions  and  the  Calculus 395 


THE  TEACHING  AND  HISTORY  OF  MATHEMATICS  IN 
THE  UNITED  STATES. 


I. 

coloinial  times. 

Elementary  Schools. 

On  tlie  study  of  mathematics  iu  elementary  schools  of  the  American 
colonies  but  little  can  be  said.  In  early  colonial  days  schools  did  not 
exist  except  in  towns  and  in  the  more  densely  settled  districts;  and 
oven  where  schools  were  kept,  the  study  of  mathematics  was  often  not 
pursued  at  all,  or  consisted  simply  in  learning  to  count  and  to  perform 
the  fundamental  operations  with  integral  numbers.  Thus,  in  Hamp- 
stead,  N.  H.,  in  1750,  it  was  voted  "  to  hire  a  school-master  for  six  mouths 
in  ye  summer  season  to  teach  ye  children  to  read  and  write."  Arith- 
metic had  not  yet  been  introduced  there.  As  late  as  the  beginning  of 
this  century  there  were  schools  in  country  districts  in  which  arithmetic 
w^as  not  taught  at  all.  Bronson  Alcott,  the  prooDinent  educator,  born  in 
Massachusetts  in  1799,  in  describing  the  schools  of  his  boyhood,  says: 
■'Until  within  a  few  years  no  studies  have  been  permitted  in  the  day 
school  but  spelling,  reading,  and  writing.  Arithmetic  was  taught  by  a 
few  instructors  one  or  two  evenings  in  a  week.  But  in  spite  of  the  most 
determined  opposition  arithmetic  is  now  permitted  in  the  day  school." 
This  was  in  Massachusetts  at  the  beginning  of  this  C(^ntury. 

In  secondary  schools,  "  ciphering"  was  taught  during  colonial  times, 
which  consisted  generally  iu  drilling  students  in  the  manipulation  of 
integral  numbers.  He  was  an  exceptional  teacher  who  possessed  a  fair 
knowledge  of  "  fractions"  and  the  "  rule  of  three,"  and  if  some  pupil  of 
rare  genius  managed  to  master  fractions,  or  even  pass  beyond  the  "  rule 
of  three,"  then  he  was  judged  a  finished  mathematician. 

The  best  teachers  of  those  days  were  college  students  or  college 
graduates  who  engaged  in  teaching  as  a  stepping-stone  to  something 
better.  An  example  of  this  class  of  teachers  was  John  Adams,  after- 
wards President  of  the  United  States.  Immediately  after  graduating 
at  Harvard  and  before  entering  upon  the  study  of  law,  he  presided,  for 
a  few  years,  over  the  grammar  scliool  at  Worcester.    From  a  letter 

9 


10  TEACHING   AND    HISTOKY    OF   MATHEMATICS. 

written  by  him  at  Worcester,  September  2,  1755,  we  clip  the  following 
description  of  the  teacher's  daily  work : 

As  a  haughty  monarch  ascends  his  throne,  the  pedagogue  mounts  his  awful  great 
chair,  and  dispenses  right  justice  through  his  whole  empire.  His  obsequious  subjects 
execute  the  imperial  mandates  with  cheerfulness,  and  think  it  their  high  happiness 
to  he  employed  in  the  service  of  the  emperor.  Sometimes  paper,  sometimes  his  pen- 
knife, now  birch,  now  arithmetic,  now  a  ferule,  then  ABC,  then  scolding,  then  flatter- 
ing, then  thwacking,  calls  for  the  pedagogue's  attention. 

School  appliances  in  those  days  were  wholly  wanting  (excepting  the 
ferule  and  birch  rods).  Slates  were  entirely  unknown  for  school  use 
until  some  years  after  the  Eevolution ;  blackboards  were  introduced 
much  later.  Paper  was  costly  in  colonial  days,  and  we  are  told  that 
birch  bark  was  sometimes  used  in  schools  in  teaching  children  to  write 
and  figure.  Thirty-six  years  ago  a  writer  in  one  of  our  magazines* 
wrote  as  follows : 

"There  are  probably  men  now  living  who  learned  to  write  on  birch 
and  beech  bark,  with  ink  made  out  of  maple  bark  and  copperas."  But 
more  generally  "  ciphering  "  was  done  on  paper.  Dr.  L.  P.  Brockett 
says  that  on  account  of  its  dearness  and  scarcity,  "the  backs  of  old  let- 
ters, the  blauk  leaves  of  ledgers  and  day-books,  and  even  the  primer 
books  were  eagerly  made  use  of  by  the  young  arithmeticians." 

Since  few  or  none  of  the  pupils  had  text-books  it  became  necessary 
for  the  teacher  to  dictate  the  "  sums."  As  in  the  colleges  of  that  time, 
so  in  elementary  schools,  tnanuscript  bools  were  used  whenever  printed 
ones  were  not  accessible.  To  advanced  boys  the  teacher  would  give 
exercises  from  his  manuscript  or  "  ciphering-book,"  in  which  the  prob. 
lems  and  their  solutions  had  been  previously  recorded.  "  With  a  book 
of  his  own  the  pupil  solved  the  problems  contained  in  it  in  their  proper 
order,  working  hard  or  taking  it  easy  as  pleased  him,  showed  the  solu- 
tions to  the  master,  and  if  found  correct  generally  copied  them  in  a  blank- 
book  provided  for  the  purpose.  *  *  *  Some  of  these  old  manuscript 
ciphering-books,  the  best,  one  may  suppose,  having  come  down  through 
several  generations,  are  still  preserved  among  old  family  records,  bear- 
ing testimony  to  the  fair  writing  and  the  careful  copying,  if  not  to  the 
arithmetical  knowledge,  of  those  who  prepared  them.  When  a  pupil  was 
unable  to  solve  a  problem  he  had  recourse  to  the  master,  who  solved  it 
for  him.  It  sometimes  happened  that  a  dozen  or  twenty  pupils  stood 
at  one  time  in  a  crowd  around  the  master's  desk  waiting  with  *  *  * 
problems  to  be  solved.  Ther^were  no  classes  in  arithmetic,  no  explana 
tions  of  processes  either  by  master  or  pupil,  no  demonstrations  of  princi- 
ples either  asked  for  or  given.  The  problems  were  solved,  the  answers 
obtained,  the  solutions  copied,  and  the  work  was  considered  complete. 
That  some  persons  did  obtain  a  good  knowledge  of  arithmetic  under 
sucli  teaching  must  be  admitted,  but  this  result  was  clearly  due  rather 
to  native  talent  or  hard  personal  labor  than  to  wise  direction."t    Those 

*  North  Carolina  University  Magazine,  Ealeigh,  1853,  Vol.  II,  p.  452. 

t  History  of  Education  in  Pennsylvania,  by  James  Pyle  Wickersham,  p.  205. 


COLONIAL  TLMES,  11 

teachers  who  were  the  fortunate  possessors  of  a,  printed  arithmetic  used 
it  as  a  guide  in  place  of  the  old  "  ciphering-book." 

In  the  early  schools,  arithmetic  was  hardly  ever  taught  to  girls.  Eev, 
William  Woodbridge  says  that  in  Connecticut,  just  before  the  Eevolu- 
tion,  he  has  "known  boys  that  could  do  something  in  the  first  four  rules 
of  arithmetic.  Girls  were  never  taught  it."*  In  the  two  "  charity 
schools "  in  Philadelphia,  which  before  the  Eevolution  were  the  most 
celebrated  schools  in  Pennsylvania,  boya  were  taught  reading,  writing, 
arithmetic;  girls,  reading,  writing,  sewing.  Thus,  sewing  was  made 
to  take  the  place  of  arithmetic.  Warren  Burton,  in  his  book  entitled, 
"  The  District  School  as  it  was,  by  one  who  went  to  it,"  says  that,  among 
girls,  arithmetic  was  neglected.  The  female  portion  "  generally  ex- 
pected to  obtain  husbands  to  perform  whatever  arithmetical  operations 
they  might  need  beyond  the  counting  of  fingers."  Occasionally  women 
were  employed  in  summer  schools  as  teachers,  but  they  did  not  teach 
arithmetic.  A  school-mistress  "would  as  soon  have  expected  to  teach 
the  Arabic  language  as  the  numerical  science^" 

The  early  school-books  in  IsTew  England  and  in  ail  other  Euglish  set- 
tlements were  much  the  same  as  those  of  Old  England.  John  Locke,  in 
his  Thoughts  concerning  Education  (1690),  says  that  the  method  of  teach- 
ing children  to  read  in  England  has  been  to  adhere  to  "  the  ordinary 
road  of  horn-book,  primer,  psalter,  testament,  and  bible."  This  same 
road  was  followed  in  New  England.  We  are  told  that  books  of  this 
kind  were  sold  to  the  people  by  John  Pynchon,  of  Springfield,  from 
1656  to  1672  and  after.!  Eegular  arithmetics  were  a  great  rarity  in 
this  country  in  the  seventeenth  century.  The  horn-book  has  been  raised 
by  some  to  the  dignified  name  of  a  "primer"  for  teaching  reading  and 
impartin  g  religious  instruction.  If  this  be  permissible,  then  why  should 
we  not  also  speak  of  it  as  an  arithmetical  primer  ?  For,  what  was  the 
horn-book?  It  consisted  of  one  sheet  of  paper  about  the  size  of  an 
ordinary  primer,  containing  a  cross  (called  "criss-cross"),  the  alphabet 
in  large  and  small  letters,  followed  by  a  small  regiment  of  monosyllables; 
then  came  a  form  of  exorcism  and  the  Lord's  Prayer,  and,  finally,  the 
Roman  numerals.  The  leaf  was  mounted  on  wood,  and  protected  with 
transparent  horn, 

"  To  save  from  fingers  wet  tbe  letters  fair." 

It  is  on  the  strength  of  the  Eoman  numerals  that  we  venture  to  pro- 
pose the  horn-book  as  a  candidate  for  the  honor  of  being  the  first  math- 
ematical primer  used  in  this  country.  Horn-books  were  quite  common 
in  England  and  in  the  English  colonies  in  America  down  to  the  time  of 
George  II.  -They  disappeared  entirely  in  this  country  before  the  Eevo- 
lution. In  early  days  the  common  remark  expressive  of  ignorance  was 
"he  does  not  know  his  horn-book,"  This  is  equivalent  to  the  more 
modern  saying,  "  he  does  not  know  his  letters." 

*  Eeminiscenses  of  Female  Education,  in  Barnard's  Journal  of  Education,  1864,  p. 
137. 
t  Barnard's  Journal  of  Education,  Vol.  XXVII. 


12  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

George  Fox,  the  founder  of  the  Society  of  Friends,  published  in  1674, 
in  England,  a  primer  or  spelling  book,  which  was  republished  at  Phila- 
delphia in  1701,  at  Boston  in  1743,  and  at  Newport,  R.  I.,  in  1769.* 
Wickersham  describes  this  little  book  as  containing  the  alphabet,  les- 
sons in  spelling  and  reading,  explanations  of  scripture  names,  Roman 
nume  als,  lessons  in  the  fundamental  rules  of  arithmetic  and  weights  and 
measures,  a  perpetual  almanac,  and  catechism  with  the  doctrine  of  the 
Friends.  It  may  be  imagined  that  a  mere  primer,  covering  such  a  wide 
range  of  subjects,  could  contain  only  a  very  few  of  the  simplest  rudi- 
ments of  a  subject  like  arithmetic.  Fox's  book  was  used  little  outside 
of  the  Society  of  Friends.^ 

Wickersham  (p.  201)  speaks  of  another  book  which  is  of  interest  as 
illustrating  the  book-making  of  those  old  times.  It  is  entitled,  "  The 
American  Instructor,  or  Young  Man's  Best  Companion,  containing  Spell- 
ing, Reading,  Writing,  Arithmetic,  in  an  Easier  Way  than  any  yet  Pub- 
lished, and  how  to  Qualify  any  Person  for  Business  without  the  Help 
of  a  Master."  It  was  written  by  George  Fisher,  and  printed  in  Phila- 
delphia, in  1748,  by  Franklin  and  Hall.  This  work  never  attained  any 
popularity. 

Dr.  Brockett  says  that  in  New  Jersey  and,  perhaps,  also  in  Virginia, 
a  book  resembling  the  "  New  England  Primer,"  but  as  intensely  Roy- 
alist and  High  Church  in  religion  as  the  New  England  Primer  was 
Puritan  and  Independent,  was  in  use  in  schools.  It  was  called  "A 
Guide  for  the  Child  and  Youth,  in  two  parts ;  the  First  for  Children,  * 
*  *  the  second  for  Youth :  Teaching  to  write,  cast  accounts  and  read 
more  perfectly;  with  several  other  varieties,  both  pleasant  and  profit- 
able. By  T.  H.,  M.  A.,  Teacher  of  a  Private  School,  London,  1762." 
It  does  not  appear  that  this  book  was  reprinted  here. 

Wickersham  gives  another  book  of  similar  stamp  but  of  much  later 
date.  "  Ludwig  Hocker's  Rechenblichiein  was  published  at  Ephrata 
[Pennsylvania]  in  1786.  The  Ephrata  publication  is  an  exceedingly 
curious  compound  of  religious  exercises  and  exercises  in  arithmetic. 
The  creed,  the  Lord's  Prayer,  hymns,  and  texts  of  scripture,  are  strangely 
intermixed  with  problems  and  calculations  in  the  simpler  parts  of  arith- 
metic"! 

One  of  the  earliest  purely  arithmetical  books  used  in  this  country  was 
the  arithmetic  of  James  Hodder.  It  may  possibly  have  fallen  into  the 
hands  of  as  early  a  teacher  as  Ezekiel  Cheever,  "the  father  of  Con- 
necticut school-masters,  the  pioneer  and  patriarch  of  elementary  classi- 
cal culture  in  New  England."!  In  a  history  of  schools  at  Salem,  Mass., 
we  are  told  that  "  among  our  earliest  arithmetics  was  James  Hodder's." 

*  History  of  Education  in  Pennsylvania.,  l>y  James  Pyle  Wickersliam,  p.  194. 

Mb\d.,  p.  200. 

t  After  having  been  a  faithful  school-master  for  seventy  years,  he  died  in  170S,  at 
the  age  of  ninety-four,  having  "held  his  abilities  in  an  unusual  degree  to  the  very 
last." 


COLONIAL    TIMES.  13 

Hodder  was  a  famous  English  teacher  of  the  seventeenth  century. 
Later  writers  have  borrowed  largely  from  his  arithmetic  of  which  the 
first  edition,  entitled  "  Hodder's  Arithmetick,  or  that  necessary  art 
made  most  easy,"  appeared  in  London  in  1661.  An  American  edition 
from  the  twenty-fifth  English  edition  was  published  in  Boston  in  1719. 
This  is  the  first  purely  arithmetical  book  known  to  have  been  printed 
in  this  country. 

In  ]^ew  York  the  Dutch  teachers  of  the  seventeenth  century  im- 
ported from  Holland  an  arithmetic  called  the  "  Coffer  Konst,"  written 
by  Pieter  Venema,  a  Dutch  school-master,  who  died  about  1612.  So 
popular  was  the  book  that  an  English  translation  of  it  was  published 
in  New  York  in  1730.  Yenema's  appeared  to  be  the  second  oldest  arith- 
metic printed  in  America. 

An  English  work  almost  as  old  as  Hodder's,  which  met  with  a  limited 
circulation  in  this  country,  is  Cocker's  Arithmetic.  The  first  edition 
appeared  in  England  after  the  death  of  Cocker,  in  1677.  According  to 
its  title  page  it  was  "  perused  and  published  by  John  Hawkins,  *  *  * 
by  the  author's  correct  copy."  De  Morgan  is  perfectly  satisfied  that 
"  Cocker's  Arithmetic  was  a  forgery  of  Hawkins's,  with  soipe  assistance, 
it  may  be,  from  Cockers  papers."  Eegarding  the  book  itself,  De  Mor- 
gan says  :*  "  Cocker's  Arithmetic  was  the  first  which  entirely  excluded 
all  demonstration  and  reasoning,  and  confined  itself  to  commercial 
questions  only.  This  was  the  secret  of  its  extensive  circulation.  There 
is  no  need  of  describing  it;  for  so  closely  have  nine  out  of  ten  of  the 
subsequent  school  treatises  been  modelled  upon  it,  that  a  large  propor- 
tion of  our  readers  would  be  able  immediately  to  turn  to  any  rule  in 
Cocker,  and  to  guess  pretty  nearly  what  they  would  find  there.  Every 
method  since  his  time  has  been  "  according  to  Cocker."  This  book  was 
found  here  and  there  in  the  colonies  at  an  early  date.  Thus  we  read 
in  Benjamin  Franklin's  Autobiography  that  (at  about  the  age  of  six- 
teen; i.  e.,  about  1722)  "  having  one  day  been  put  to  the  blush  for  my 
ignorance  in  the  art  of  calculation  which  I  had  twice  failed  to  learn 
while  at  school,  I  took  Cocker's  trei^tise  on  arithmetic  and  went  througii 
it  by  myself  with  the  utmost  ease."  An  American  edition  of  the  work 
appeared  in  Philadelphia  in  1779.  It  contains  the  rude  portrait  of  the 
author,  "  which  might  be  taken  for  a  caricature,"  and  also  the  following 
poetical  recommendation : 

Ingenious  Cocker,  now  to  Rest  thou  'rt  gone, 
No  Art  can  show  thee  fully,  but  thine  own; 
Thy  rare  Arithmetick  alone  can  show 
Th'  vast  Thanks  we  for  thy  labours  owe. 

Wickershamt  mentions  Baniel  Yenmng^s  Ber  Oesohwinde  Rechner  as 
having  been  published  by  Sower  in  1774. 


*  Article,  "  Cocker,"  Penny  Cyclopaedia, 

t  History  of  Education  in  Pennsylvania,  p.  200. 


14  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

The  first  arithmetic  written  by  an  American  author  and  printed  here 
was  that  of  Prof.  Isaac  Greenwood  of  Harvard  College,  in  1729.  The 
book  was  probably  used  by  the  author  in  his  classes  at  Harvard.  We 
have  nowhere  seen  it  mentioned  except  in  a  biographical  sketch  of  its 
author.*  So  far  as  we  know,  there  are  only  three  copies  of  Greenwood's 
Arithmetic  in  existence,  two  in  the  Harvard  library  and  one  in  the  Con. 
gressional  Library.  Prof.  J.  M.  Greenwood,  superintendent  of  schools 
in  Kansas  City,  sends  the  writer  the  following  descrij)tion  of  it : 

The  book  is  a  small  duodecimo  volume  of  158  pages^  exclusive  of  an 
advertisement  (4  pages)  prefixed,  and  the  table  of  contents  (4  pages) 
put  at  the  end.  The  following  is  a  transcript  of  the  title-page :  "Arith- 
metick,  Vulgar  and  Decimal:  with  the  Application  thereof  to  a  variety 
of  Cases  in  Trade  and  Commerce.  (Vignette.)  Boston :  N.  E.,  Printed 
by  S.  Kneeland  and  T.  Green,  for  T.  Hancock  at  the  Sign  of  the  Bible 
and  Three  Crowns,  in  Ann  Street,  MDCCXXIX." 

The  headings  of  chapters  are  as  follows :  The  introduction ;  chapter 
1,  Numeration ;  chapter  2,  Addition  j  chapter  3,  Subtraction ;  chapter 
4,  Multiplication  5  chapter  5,  Division ;  chapter  6,  Eeduction ;  chapter 
7,  Vulgar  FiBctions ;  chapter  8,  Decimal  Fractions;  chapter  9,  Eoots 
and  Powers;  chapter  10,  Continued  Proportion;  chapter  11,  Disjunct 
Proportion ;  chapter  12,  Practice;  chapter  13,  Eales  relating  to  Trade 
and  Commerce. 

From  the  preface :  "The  Author's  Design  in  the  following  Treatise  is  to  give  a  very- 
concise  Account  of  such  Rules,  as  are  of  the  easiest  practice  in  all  the  Parts  of  Vulgar 
and  Decimal  Arithmfetick  and  to  illustrate  each  with  such  esamples,  as  may  be  suf- 
ficient to  lead  the  Learner  to  the  full  Use  thereof  in  all  other  Instances." 

"  The  Reader  will  observe  that  the  Author  has  inserted  under  all  those  Rules,  where 
it  was  proper,  Examples  with  Blanks  for  his  Practice.  This  was  a  Principal  End  to 
the  Undertaking ;  that  such  persons  as  were  desirous  thereof  might  have  a  compre- 
hensive Collection  of  all  the  best  Rules  in  the  Art  of  Numbering,  with  Examples 
wrought  by  themselves.  And  that  nothing  might  be,.wanting  to  favour  this  Design, 
the  Impression  is  made  upon  several  of  the  best  sorts  of  Paper.  This  method  is  en- 
tirely new,    *    *    *." 

The  paper  used  in  the  book  is  thick,  the  type  large.  Words  and 
phrases  to  which  the  author  desires  to  call  special  attention  are  printed 
in  italic  characters,  and  as  more  than  half  the  book  is,  in  the  author's 
eyes,  important,  more  than  half  the  book  is  printed  in  italics. 

In  1788,  when  Nicholas  Pike  published  his  arithmetic.  Greenwood's 
book  was  entirely  unknown,  and  Pike's  was  believed  to  be  the  first  arith- 
metic written  and  i)rinted  in  America. 

The  first  arithmetic  which  enjoyed  general  popularity  and  reached  an 
extended  circulation  in  the  colonies  was  the  School-master's  Assistant, 
by  Thomas  Dilworth.  The  first  edition  of  this  was  published  in  London 
in  1744  or  '45.  According  to  Wickersham,  there  appeared  a  reprint  of 
this  in  Philadelphia  in  1769.  Other  American  editions  were  brought 
out  at  Hartford  in  1786,  New  York  in  1793  and  1806,  Brooklyn  in  1807, 

*  Apjsleton's  Dictionary  of  American  Biography. 


COLONIAL   TIMES.  15 

Hew  Londoii  1797,  aud  Albany  1824.  At  the  beginning  of  tbe  Eevolo- 
tion  this  was  the  most  popular  arithmetic,  and  it  continued  in  use  long 
after, 

We  have  now  enumerated  all  the  arithmetics  which  were  used  to  our 
knowledge  in  the  American  colonies.  It  may  be  instructive  to  give  the 
last  book  which  we  have  mentioned  a  closer  examination  ;  for  Dil worth's 
School-master's  Assistant  was  the  most  noted  arithmetic  of  its  time.  As 
an  arithmetician  Dilworth  belonged  to  the  school  founded  by  Cocker, 
which  scrupulously  excluded  all  demonstration  and  reasoning.  The 
School-master's  Assistant  gives  all  rules  and  definitions  in  the  form  of 
questions  and  answers.  Let  us  turn  to  page  M  of  the  twenty- second 
London  edition,  1784,  and  examine  his  mode  of  explaining  proportion, 
or,  as  the  subject  was  then  called,  the  "  Eule  of  Three," 

OF  THE  SINGLE  RULE  OF  THKEB. 

Q.  How  many  Parts  are  tliera  in  the  Eule  of  Three  ? 

A.  Two :  Single  or  Simple,  and  Double  or  Compound. 

Q.  By  what  is  the  single  Eule  of  Three  known  ? 

A.  By  three  Terms,  which  are  always  given  in  the  Question,  to  find'a  fourth. 

Q.  Ave  any  of  the  terms  given  to  be  reduced  from  one  Denomination  to  another? 

A.  If  any  of  the  given  terms  be  of  several  denominations,  they  must'  tse  reduced  into 

the  lowest  Denomination  mentioned.  ^ 

Q.  What  do  you  observe  concerning  the  first  and  third  Terms  I 
A.  They  must  be  of  the  same  Name  and  Kind. 
Q.  What  do  you  observe  concerning  the  fourth  Term  ?  ■ 
A.  It  must  be  of  the  same  Name  and  Kind  as  the  second. 
Q.  What  do  you  observe  of  the  three  given  Terms  taken  togethel?f~ 
A.  That  the  two  first  are  a  Supposition,  the'last  is  a  Demand. ' 
Q.  How  is  the  third  Term  known  ?  ♦ 

A.  It  is  known  by  these,  or  the  like  Words,  What  cost  ?   Hoio  many  ?    Eoxv  much  f 
Q.  How  many  Sorts  of  Proportion  are  there  ? 
A.  Two:  Direct  and  Inverse. 

And  so  on.  We  have  quoted  enough  to  give  an  idea  of  the  book.  It 
is  not  easy  to  see  how  a  pupil  beginning  the  subject  of  proportion  could 
get  clear  notions  from  reading  the  above.  Kor  can  we  see  how  a  boy 
who  had  never  before  heard  of  fractions  could  get  any  idea  whpvtever 
of  a  fraction  from  Dilworth's  definition,  which  is  (p.  Ill) :  A  fraction 
"  is  a  broken  number  and  signifies  the  part  or  parts  of  a  whole  num- 
ber." 

A  closer  examination  of  this  arithmetic  discloses  many  other  strange 
things.  It  consists  really  of  three  parts,  more  or  less  complete  in  them- 
selves, namely:  Parti,  on  whole  numbers ;  Part  II,  on  vulgar  fractions; 
Part  III,  on  decimal  fractions.  In  Part  I,  the  student  is  carried  through 
the  elementary  rules,  and  through  interest,  fellowship,  exchange,  double 
rule  01  three,  aliigation,  single  and  double  position,  geometrical  pro- 
gression, and  permutations.  He  is  carried  through  all  these  without 
having  as  yet  even  heard  effractions.  The  advanced  and  comparatively 
unimportant  subjects,  such  as  ailigation  and  progressionsj  are  mad© 


16  TEACHING    AND   HISTOEY    OP   MATHEMATICS. 

to  precede  so  important  and  fundamental  a  subject  as  fractions.  The 
teaching  of  decimal  fractions  after  interest  is  illogical,  to  say  the  least. 
In  Part  II,  after  fractions  have  been  explained,  the  rule  of  three  is  taken 
up  a  second  time';  and  in  Part  III,  under  decimal  fractions,  it  is  re- 
sumed a  third  time.  Thus  this  rule  is  exi^lained  three  times ;  the  first 
time  with  whole  numbers,  the  second  time  with  common  fractious,  the 
third  time  with  decimal  fractions — thus  leaving  the  impression  that  the 
rule  is  different  in  each  one  of  the  three  cases. 

The  whole  book  is  nothing  but  a  Pandora's  box  of  disconnected  rules. 
It  appeals  to  memory  exclusively  and  completely  ignores  the  existence 
of  reasoning  powers  in  the  mind  of  the  learner.  Koticeable  is  the  fact 
that  in  the  treatment  of  common  fractions,  the  process  of"  cancellation," 
which  may  be  made  to  shorten  operations  so  much,  is  not  even  men- 
tioned. The  book  abounds  in  unnecessary  and  perplexing  technical 
terms,  such  as  "  practice,"  '^  conjoined,  proportion,"  "alligation  medial," 
<' alligation  alternate,"  "  comparative  arithmetic,"  "biquadrate  roots," 
"sursolids,"  "square  cubes,"  "secontlsursolids,"  "biquadrates  squared," 
"third  sursolids,"  and  "square  cubes  squared."  Under  the  head  of 
duodecimals  are  given  rules  like  these:  "  Feet  multiplied  by  feet  give 
feet; "  "feet  multiplied  by  inches  give  inches,"  etc.  Tliese  rules,  taken 
literally,  are  absurd.  We  can  no  more  multiply  feet  by  feet  than  we 
can  multiply  umbrellas  by  umbrellas.  These  rules  are  in  opposition  to 
the  fundamental  ideas  of  multiplication  in  arithmetic.  A  concrete 
number  can  not  be  multiplied  by  a  concrete  number.  It  seems  strange 
that  so  gross  a^  error  should  not  have  been  corrected  in  later  editions 
of  the  book ;  but  still  more  strange  is  the  fact  that  nearly  all  arithme- 
tics down  to  the  i^resent  day  should  have  persisted  in  making  this 
mistake. 

As  an  instance  of  the  confusion  of  ideas  to  which  it  gives  rise,  I  quote 
the  following  from  an  article  "Early  School  Days "  in  Indiana,  con- 
tributed by  Barnabas  C.  Hobbs.*  A  law  had  just  been  passed  requir- 
ing that  teachers'  examinations  should  be  conducted  by  three  county 
examiners  instead  of  the  township  trustees,  as  had  been  the  practice 
before.  "I  shall  not  forget,"  says  Hobbs,  'imy  first  experience  under 
the  new  system.  The  only  question  asked  me  at  my  first  examination 
■was,  'What  is  the  product  of  25  cents  by  25  cents T  *  *  *  We 
were  not  as  exact  then  as  people  are  now.  We  had  only  Pike's  Arith- 
metic, which  gave  the  sums  and  the  rules.  These  were  considered 
enough  at  that  day.  How  could  I  tell  the  product  of  25  cents  by  25 
cents,  when  such  a  problem  could  not  be  found  in  the  book  ?  The  ex- 
aminer thought  it  was  6i-  cents,  but  was  not  sure.  I  thought  just  as  he 
did,  but  this  looked  too  small  to  both  of  us.  We  discussed  its  merits 
for  an  hour  or  more,  when  he  decided  that  he  was  sure  I  was  qualified 
to  teach  school,  and  a  first-class  certificate  was  given  me." 

*  Tfie  Indiana  Schools,  by  James  H.  Smart,  1876. 


COLONIAL    TIMES.  17 

We  have  spoken  of  Dilwortli's  School-master's  Assistant  at  some 
length,  because  from  it  we  caa  see  what  sort  of  arithmetics  we  inherited 
from  the  English.  All  arithmetics  of  that  time  were  much  alike.  The 
criticisms  upon  one  will  therefore  apply  to  all. 

Before  proceeding  to  another  subject  we  shall  examine  briefly  the 
'•  Short  Collection  of  Pleasant  and  Diverting  Questions  "  in  Dilworth. 
We  shall  meet  there  with  a  company  of  familiar  friends.  Who  has  not 
heard  of  the  farmer,  who,  having  a  fos,  a  goose,  and  a  peck  of  corn, 
and  wishing  to  cross  a  river,  but  being  able  to  carry  but  one  at  a  ticne, 
was  confounded  as  to  how  he  should  carry  them  across  so  that  the  fox 
should  not  devour  the  goose,  nor  the  goose  the  corn?  Who  has  not  heard 
of  the  perplexing  problem  of  how  three  jealous  husbands  with  their 
wives  may  cross  a  river  in  a  boat  holding  only  two,  so  that  none  of  the 
three  wives  shall  be  found  in  company  of  one  or  two  men,  unless  her  hus- 
band be  present  ?  Many  of  us,  no  doubt,  have  also  been  asked  to  place 
the  nine  digits  in  a  quadrangular  form  in  such  a  way  that  any  three  fig- 
ures in  a  line  may  make  just  15?  When  these  pleasing  problems  were 
first  proposed  to  us,  they  came  like  the  mofning  breeze,  with  exhilarat- 
ing freshness.  We  little  suspected  that  these  apparently  new-born 
creatures  of  fancy  were  in  reality  of  considerable  antiquity;  that  they 
were  found  in  an  arithmetic  used  in  this  country  one  hundred  years  ago. 
Still  greater  is  our  surprise  when  we  learn  that  at  the  time  they  were 
published  in  Dilworth's  School-master's  Assistant  some  of  these  ques- 
tions for  amusement  had  already  seen  as  many  as  one  thousand  birth- 
days. The  oldest  record  bearing  upon  this  subject  is  found  in  a  manu- 
script entitled  Propositioncs  ad  acuendos  jtivenes.  The  authorship  of 
this  paper  has  been  generally  attributed  to  Alcuin,  whose  years  of  great- 
est activity  were  spent  in  France,  in  the  court  of  the  great  Charlemagne, 
and  who  was  one  of  the  most  learned  scholars  and  celebrated  teachers 
of  the  eighth  century.  The  MS.  attributed  to  him  contains  the  puzzle 
about  the  wolf,  goat,  and  cabbage,  which  in  the  modern  version  is  known 
as  the  "fox,  goose,  and  peck  of  corn"  puzzle. 

In  a  MS.  coming  from  the  thirteenth  century,  two  learned  German 
youths,  named  Firri  and  Tyrri,  are  made  to  propose  to  each  other  prob- 
lems and  puzzles.  Firri  takes  among  others  the  hard  nut  of  Alcuin 
about  the  wolf,-goat,  and  cabbage-head,  and  lays  it  before  Tyrri  in  the 
modified  and  improved  version  of  the  three  wives  and  the  three  jealous 
husbands.  This  same  document  contains  also  the  following:  "Firri 
says :  There  were  three  brothers  in  Cologne,  having  nine  vessels  of  wine. 
The  first  vessel  contained  1  quart  (amam),  the  second  2,  the  third  3,  the 
fourth  4,  the  fifth  5,  the  sixth  6,  the  seventh  7,  the  eighth  8,  the  ninth  9. 
Divide  the  wine  equally  among  the  three  brothers,  without  mixing  the 
contents  of  the  vessels." 

This  problem  admits  of  more  than  one  solution,  and  is  closely  related 
to  the  last  problem  we  quoted  from  Diiworth's  collection.  It  is  of  spe- 
881— Xo.  3 2 


18 


TEACHING   AND   HISTORY    OF    MATHEMATICS. 


cial  interest,  since  it  gives  rise  to  the  follovring  magic  square,  in  which 
any  three  figures  in  a  straight  line  have  15  for  their  sum. 


2 
9 

7 
5 

6 

I 

4 

3 

8 

The  history  of  magic  squares  is  a  rich  field  for  investig'ation.  The 
Germans  were  by  no  means  the  originators  of  them.  This  honor  must 
be  given  to  the  Brahmins  in  India.  Later  on  the  study  of  these  curi- 
ous problems  was  zealously  pursued  by  the  Arabs,  who  transmitted  the 
fruits  of  their  study  to  the  Europeans. 

Had  we  the  time,  we  would  attempt  to  trace  the  history  of  some 
other  familiar  puzzles.  But  enough  has  been  said  to  show  that  many 
of  them  possess  great  antiquity.  Nevertheless,  when  they  were  first 
proposed  to  us,  they  betrayed  no  signs  of  old  age.  May  they  continue 
perjjetually  in  their  youth,  and  may  they  delight  the  minds  of  men 
for  numberless  centuries  to  come  ! 

Colleges, 
haryakd  college. 

As  early  as  1636  the  people  of  Massachusetts  stamped  their  approval 
upon  the  cause  of  higher  education  by  the  founding  of  Harvard  College. 
The  nature  of  the  early  instruction  given  at  this  oldest  of  American 
colleges  is  of  special  interest  to  us.  The  earliest  record  bearing  on  the 
history  of  the  rise  of  mathematical  studies  at  Harvard  is  a  tract  en- 
titled, ''  New  England's  First  Fruits."  It  was  originally  published  in 
1643,  or  five  years  after  the  college  had  opened,  and  contained  the  cur- 
riculum of  studies  then  pursued.  Whoever  expects  to  find  in  it  an  ex- 
tended course  of  mathematical  studies  resembling  that  in  our  colleges 
of  to-day  will  be  much  disappointed. 

In  the  first  place,  a  student  applying  for  admission  to  Harvard  in 
1643  was  not  confronted  and  embarrassed  by  any  entrance  examinations 
in  mathematics.  The  main  requirement  for  admission  was  Latin.  Con- 
trary to  the  practice  of  to-day,  Latin  was  then  taught  as  a  spoken  lan- 
guage. "  So  much  Latin  as  was  sufficient  to  understand  Tally,  or  any 
like  classical  author,  and  to  make  and  speak  true  Latin  in  prose  or  verse, 
and  so  much  Greek  as  was  included  in  declining  perfectly  the  para- 
digms of  the  Greek  nouns  and  verbs,"  \vere  the  necessary  requisites  for 
admission ;  but  in  mathematics  applicants  were  required  to  know  not 
even  the  multiplication  table. 

When  we  come  to  examine  the  college  course,  which  extended  origi- 
nally through  only  three  years,  we  meet  with  other  surprises.     Boys  did 


COLONIAL    TIMES.  .19 

not  receive  that  thorough  "  grinding"  in  the  elements  during  the  first 
years  ot  college  that  they  do  now ;  on  the  contrary,  no  mathematics  at 
all  was  taught  except  during  the  last  year.  The  mathematical  course 
began  in  the  Senior  year,  and  consisted  of  arithmetic  and  geometry 
during  the  iirst  three-quarters  of  the  year,  and  astronomy  during  the 
last  quarter.  Algebra  was  then  an  unknown  science  in  the  New  World. 
It  is  interesting  to  notice  that,  in  this  original  curriculum,  the  atten- 
tion of  each  class  was  concentrated  for  a  whole  day  upon  only  on6  or 
two  subjects.  Thus,  Mondays  and  Tuesdays  were  devoted  by  the  third 
year  students  exclusively  to  mathematics  or  astronomy,  Wednesdays  to 
Greek,  Thursdays  to  "Eastern  tongues."  and  so  on.  The  importance 
attached  to  mathematical  studies,  as  compared  with  other  branches  of 
discipline,  may  be  inferred  from  the  fact  that  ten  hours  per  week  were 
devoted  to  philosophy,  seven,  to  Greek,  sis  to  Ehetoric,  four  to  Oriental 
languages,  but  only  two  to  mathematics.  According  to  these  figures, 
Oriental  languages  were  considered  twice  as  important  as  mathematics. 
But  we  must  remember  that  this  course  was  laid  out  for  students  who 
were  supposed  to  choose  the  clerical  profession.  For  that  reason,  phil- 
osophical, linguistic,  and  theological  studies  were  allowed  to  monopolize 
nearly  the  whole  time,  while  mathematics  was  excluded  almost  en- 
tirely. 

In  what  precedes  we  have  measured  the  college  work  done  in  1643 
by  the  standards  of  1889.  Let  us  now  compare  it  with  the  contempo- 
raneous work  in  English  universities.  We  may  here  premise  that  in  the 
middle  of  the  seventeenth  century  rapid  progress  was  made  in  the 
mathematical  sciences.  In  1643,  Galileo  had  just  passed  away  j  Cav- 
alieri,  Torricelli,  Pascal,  Format,  Eoberval,  and  Descartes  were  at  the 
zenith  of  their  scientific  activity;  John  Wallis  was  a  young  man  of 
twenty-seven,  Isaac  Barrow  a  youth  of  thirteen,  while  Isaac  Newton 
was  an  infant  feeding  from  his  mother's  breast.  Though  much  original 
work  was  being  done,  especially  by  French  and  Italian  mathematicians, 
the  enthusiasm  for  mathematical  study  had  hardly  reached  the  univer- 
sities. Some  idea  of  the  state  of  mathematics  at  Cambridge,  England, 
previous  to  the  appearance  of  Newton,  may  be  gathered  from  a  dis- 
cburse by  Isaac  Barrow,  delivered  in  Latin,  probably  in  1654,  or 
eighteen  years  after  the  founding  of  Harvard  College.  In  it  occurs  the 
following  passage:  "The  once  horrid  names  of  Euclid,  Archimedes, 
Ptolemy,  and  Diophantus,  many  of  us  no  longer  hear  with  trembling 
ears.  Why  should  I  mention  the  fact  that  by  the  aid  of  arithmetic,  we 
have  now  learned,  with  easy  and  instantaneous  work,  to  compute  ac- 
curately the  number  of  the  very  sands  (themselves).  *  *  *  And  in- 
deed that  horrible  monster  that  men  call  algebra  many  of  us  brave  men 
(that  we  are)  have  overcome,  put  to  flight,  and  (fairly)  triumphed  over ; 
(while)  very  many  (of  us)  have  dared,  with  straight-along  glance,  to 
look  into  optics;  and  others  (still),  with  intellectual  rays  unbroken, 
have  dared  to  pierce  (their  way)  into  the  still  subtler  and  highly  useful 
doctrine  of  dioptrics." 


20  TEACHING    AND   HISTORY    OF    MATHEMATICS. 

From  this  it  would  seem  that  mathematical  studies  had  been  intro- 
duced into  old  Cambridge  only  a  short  time  before  Barrow  deli%'ered 
his  speech.  It  thus  appears  that  about  1636,  when  new  Cambridge  was 
founded  in  the  wilds  of  the  west,  old  Cambridge  was  not  mathematical 
at  all.  In  further  support  of  this  view  we  quote  from  the  Penny  Cy- 
clopedia, article  "  Wallis,"  the  following  statement:  ''There  were  no 
mathematical  studies  at  that  time  [when  Wallis  entered  Emmanuel 
College  in  1632]  at  Cambridge,  and  none  to  give  even  so  much  as  advice 
what  books  to  read.  The  best  mathematicians  were  in  London,  and  the 
science  was  esteemed  no  better  than  mechanical.  This  account  is  con. 
firmed  by  his  [Wallis's]  contemporary,  Horrocks,  who  was  also  at  Em- 
manuel and  whose  works  Wallis  afterwards  edited."  In  a  biography  of 
Seth  Ward,  an  Euglish  divine  and  astronomer,  we  meet  with  similar 
testimony.*  He  entered  Sidney  Sussex  College,  Cambridge,  in  1632. 
"  In  the  college  library  he  found,  by  chance,  some  books  that  treated 
of  the  mathematics,  and  they  being  wholly  new  to  him,  he  inquired  all 
the  college  over  for  a  guide  to  instruct  him  that  way,  but  all  his  search 
was  in  vain ;  these  books  were  Greek,  I  mean  unintelligible,  to  all  the 
fellows  in  the  college." 

If  so  little  was  done  at  old  Cambridge,  then  we  need  not  wonder  at 
the  fact  that  new  Cambridge  failed  to  be  mathematical  from  the  start. 
The  fountain  could  not  rise  higher  than  its  source.  It  was  not  until  the' 
latter  half  of  the  seventeenth  century  that  mathematical  studies  at 
old  Cambridge  rose  into  prominence.  Impelled  by  the  genius  of  Sir 
Isaac  IS^ewtou,  old  Cambridge  advanced  with  such  rapid  strides  that 
the  youthful  college  in  the  west  became  almost  invisible  in  the  distant 
rear. 

The  mathematical  course  at  Harvard  remained  apparently  the  same 
till  the  beginning  of  the  eighteenth  century.  Arithmetic  and  a  little 
geometry  and  astronomy  constituted  the  sum  total  of  the  college  in- 
struction in  the  exact  sciences.  Applicants  for  the  master's  degree 
had  only  to  go  over  the  same  ground  more  thoroughly.  Says  Cotton 
Mather:  "Every  scholar  that  giveth  up  in  writing  a  system  or  synop- 
sis or  sum  of  logic,  natural  and  moral  philosophy,  arithmetic,  geometry, 
and  astronomy,  and  is  ready  to  defend  his  theses  or  positions,  withal 
skilled  in  the  originals,  as  above  said,  and  of  godly  life  *  *  *  is  fit 
to  be  dignified  with  the  second  degree." t 

These  few  unsatisfactory  data  are  the  only  fragments  of  information 
that  we  could  find  on  the  mathematical  course  at  Harvard  during  the 
seventeenth  century.  The  following  note  on  the  nature  of  the  instruc- 
tion given  in  physics  is  not  without  interest:  Mr.  Abraham  Pierson, 
jr.  (first  rector  of  Yale  College),  graduated  at  Harvard  in  16G8.  The 
college  (Yale)  possesses  several  of  his  MSS., "  containing  notes  made  by 

*  Life  of  Eigbt  Reverend  Seth,  Lord  Bishop  of  SaJisbury,  by  Walter  Pope.    Lou- 
don, 1697,  p.  9. 
t  Magnalia,  Book  IV,  128th  ed,,  1702. 


COLONIAL    TIMES.  21 

him  during  his  student  life  at  Harvard  on  logic,  theology,  and  physics, 
and  so  throwing  light  on  the  probable  compass  of  the  manuscript  text- 
book on  physics  compiled  by  him,  which  was  handed  down  from  one 
college  generation  to  another  for  some  twenty-five  years,  until  super- 
seded by  Clarke's  Latin  translation  of  Eohault's  Traite  de  Fhysiqiie.  * 
The  Harvard  notes  on  physics  seem  (from  an  inscription  attached)  to 
to  have  been  derived  in  like  manner  from  the  teachings  of  the  Eev. 
Jonathan  Mitchel  (Harvard  College,  1647);  they  are  rather  metaphysi- 
cal than  mathematical  in  form,  and  it  is  even  difftcult  to  determine 
what  theories  of  physical  astronomy  the  writer  held.  Suffice  it  to  say 
that  he  ranged  himself  somewhere  in  the  wide  interval  between  the 
Ptolemaic  theory  (generally  abandoned  one  hundred  years  earlier)  and 
the  Newtonian  theory  (hardly  known  to  any  one  in  this  part  of  the  world 
until  the  eighteenth  century).  In  other  words,  while  recognizing  that 
the  earth  is  round,  and  that  there  is  such  a  force  as  gravity,  there  is  no 
proof  that  he  had  got  beyond  Copernicus  to  Kepler  and  G-alileo."  * 

In  this  extract  our  attention  is  also  called  to  the  common  practice 
among  successive  generations  of  students  at  that  time  of  copying  manu. 
script  text-books.  As  another  instance  of  this  we  mention  the  manu- 
script works,  a  System  of  Logic  and  a  Compendium  Phj^sical,  by  Eev. 
Charles  Morton,  which  (about  1692)  were  received  as  text-books  at 
Harvard,  "  the  students  being  required  to  copy  them."t 

We  shall  frequently  have  occasion  to  observe  that  astronomical  pur- 
suits have  always  been  followed  with  zeal  and  held  in  high  estimation 
by  the  American  people.  As  early  as  1651  a  l^ew  England  writer,  in 
naming  the  "  first  fruits  of  the  college,"  speaks  of  the  "  godly  Mr.  Sam 
Danforth,  who  hath  not  only  studied  divinity,  but  also  astronomy ;  he 
put  forth  many  almanacs,"  and  "  was  one  of  the  fellows  of  the  college." 
Another  fellow  of  Harvard  was  John  Sherman.  He  was  a  popular 
preacher,  an  "  eminent  mathematician,"  and  delivered  lectures  at  the 
college  for  many  years.  He  published  several  almanacs,  to  which  he 
appended  pious  retlections.  The  ability  of  making  almanacs  was  then 
considered  proof  of  profound  erudition.  A  somewhat  stronger  evidence 
of  the  interest  taken  in  astronomy  was  the  publication  at  Cambridge 
of  a  set  of  astronomical  calculations  by  Uriah  Oakes.  Oakes,  at  that 
time  a  young  man,  had  graduated  at  Harvard  in  1049,  and  in  1680  be- 
came president  irro  tern,  and  afterwards  president  of  Harvard  College, 
In  allusion  to  his  size,  he  attached  to  his  calculations  the  motto, 
^^ Farvum  ;parva  decent^  sed  inest  sua  gratia  'par vis. ^^  (Small  things  befit 
the  small,  yet  have  a  charm  their  own.) 

The  preceding  is  an  account  of  the  mathematical  and  physical  studies 
at  Harvard  during  the  seventeenth  century.  We  now  proceed  to  the 
eighteenth  century.     It  appears  that  in  1700  algebra  had  not  yet  be- 

*  Yale  Biographies  and  Annals,  1701-1745,  by  Franklin  Bowditch  Dexter,  p.  61. 
tQuincy's  History  of  Harvard  University,  Vol.  1,  p.  70. 


22  TEACHING   AND  HISTORY    OF    MATHEMATICS. 

coine  a  college  study.  The  Autobiography  of  Eev.  John  Barnard  * 
throws  some  light  oil  this  subject.  Barnard  took  his  first  degree  at 
Harvard  in  1700,  then  returned  to  his  father's  house,  where  he  betook 
hiiQself  to  studying.  "  While  I  continued  at  my  father's  I  prosecuted 
'*my  studies  and  looked  something  into  the  mathematics,  though  I  gained 
but  little,  our  advantages  therefor  being  noways  equal  to  what  they 
have  who  now  have  the  great  Sir  Isaac  Xewton  and  Dr.  Halley  and 
some  other  mathematicians  for  their  guides.  About  this  time  I  made 
a  visit  to  the  college,  as  I  generally  did  once  or  twice  a  year,  where  I 
remember  the  conversation  turning  upon  the  mathematics,  one  of  the 
company,  who  was  a  considerable  proficient  in  them,  observing  my  ig- 
norance, said  to  me  he  would  give  me  a  question,  which  if  I  answered 
in  a  month's  close  application  he  should  account  me  an  apt  scholar. 
He  gave  me  the  question.  I,  who  was  ashamed  of  the  reproach  cast 
upon  me,  set  myself  hard  to  work,  and  in  a  fortnight's  time  returned 
him  a  solution  of  the  question,  both  by  trigonometry  and  geometry, 
with  a  canon  by  which  to  resolve  all  questions  of  the  like  nature. 
When  I  showed  it  to  him  he  was  surprised,  said  it  was  right,  and 
owned  he  knew  no  other  way  of  resolving  it  but  by  algebra,  which  I 
was  an  utter  stranger  to."  Though  a  graduate  of  Harvard,  he  was  an 
utter  stranger  to  algebra.  From  this  we  may  safely  conclude  that  in 
1700  algebra  was  not  yet  a  part  of  the  college  curriculum. 

What,  then,  constituted  the  mathematical  instruction  at  that  time? 
Was  it  any  diiferent  from  the  course  given  in  1643  ?  Until  about  1655, 
the  entire  college  course  extended  through  only  three  years ;  at  this 
time  it  was  lengthened  to  four  years.  We  might  have  supposed  that 
the  mathematics  formerly  taught  in  the  third  year  would  have  been 
retained  as  a  study  for  the  third  or  Junior  year,  but  this  was  not  the 
case.  In  the  four-years'  course,  mathematics  was  taught  during  the 
last,  or  Senior  year.  Quincy,  in  his  history  of  Harvard  University  (Vol. 
I,  p.  441),  quotes  from  Wad  worth's  Diary  the  list  of  studies  for  the  year 
1726.  The  Freshmen  recited  in  Tully,  Virgil,  Greek  testament,  rheto- 
ric, Greek  catechism;  the  Sophomores  in  logic,  natural  philosophy, 
classic  authors,  Heerebord's  Meletemata,  Wollebins's  Divinity;  the 
"  Junior  sophisters"  in  Heerebord's  Meletemata,  physics,  ethics,  geogra- 
phy, metaphysics;  while  the  "Senior  sophisters,  besides  arithmetic, 
recite  Alsted's  Geometry,  Gassendi's  Astronomy  in  the  morning;  go 
over  the  arts  towards  the  latter  end  of  the  year,  Ames's  Medulla  on 
Saturdays,  and  dispute  once  a  week."  This  quotation  establishes  the 
fact  that  ninety  years  after  the  founding  of  Harvard,  the  mathematical 
course  was  essentially  the  same  as  at  the  beginning.  Arithmetic, 
geometry,  and  astronomy  still  constituted  the  entire  course.  Mathe- 
matics continued  to  be  considered  the  crowning  pinnacle  instead  of  a 
corner-stone  of  college  education ;  natural  philosophy  and  physics  were 

.     *  Collec/iious  of  the  Mdss.  Hist.  Soc,  Third  Series,  Vol.  V,  pp.  177-243. 


COLONIAL    TIMES.  23 

still  taught  before  arithmetic  aud  geometry.  But  we  must  observe  that, 
iu  1726,  j;H?iie(^  treatises  were  used  as  test-books  in  geometry  and  astron- 
omy. We  are  not  informed  at  what  time  these  printed  books  were  in- 
troduced. They  may  have  been  used  as  text-books  much  earlier  than 
the  above  date.  The  authors  of  these  books  were  in  their  day  scholars 
of  wide  reputation.  Johann  Heinrich  Alsted  (1558-1638),  the  author  of 
the  Geometry,  was  a  German  Protestant  divine,  a  professor  of  philoso- 
phy and  divinity  at  Herborn  in  Nassau,  and  afterwards  in  Oarlsburg  in 
Transylvania.  In  one  of  his  books  he  maintained  that  the  millenium 
was  to  come  in  1694. 

Pierre  Gassendi  (1592-1655),  whose  little  astronomy  of  one  hundred  and 
fifty  Images  was  used  as  a  class-book  at  Harvard,  was  a  contemporary  of 
Descartes  and  one  of  the  most  distinguished  naturalists,  mathematicians, 
and  philosophers  of  France.  He  w^as  for  a  time  professor  of  mathe- 
matics at  the  College  Eoyal  of  Paris.  What  seems  very  strange  to  us 
is  that  nearly  a  century  after  the  first  publication  of  these  books  they 
should  have  been  still  in  use  and  apparently  looked  upon  as  the  best  of 
'their  kind.  Forty  years  after  the  publication  of  ISTewton's  Principia 
an  astronomy  was  being  studied  at  Harvard  whose  author  died  before 
the  name  of  ]!^ewton  had  become  known  to  science.  The  wide  chasm 
between  the  theories  of  J^ewton  and  those  of  Gassendi  is  brought  to  full 
view  by  the  following  quotation  from  Whewell's  History  of  the  Induc- 
tive Sciences  (Third  edition,  Yol.  I,  p.  392) :  "  Gassendi's  own  views  of 
the  causes  of  the  motions  of  the  heavenly  bodies  are  not  very  clear. 
*  *  *  In  a  chapter  headed  'Quae  sit  motrix  siderum  causa,'  he 
reviews  several  opinions ;  but  the  one  which  he  seems  to  adopt  is  that 
which  ascribes  the  motion  of  the  celestial  globes  to  certain  fibers, 
of  which  the  action  is  similar  to  that  of  the  muscles  of  animals.  It 
does  not  ajjpear  therefore  that  he  had  distinctly  apprehended,  either 
the  continuation  of  the  movements  of  the  planets  by  the  first  law  of 
motion,  or  their  deflection  by  the  second  law." 

The  year  1726  is  memorable  in  the  annals  of  Harvard  for  the  estab- 
lishing of  the  Hollis  professorship  of  mathematics.  Thomas  Hollis, 
a  kind-hearted  friend  of  the  college,  transmitted  to  the  treasurer  of  the 
college  the  then  munificent  sum  of  twelve  hundred  pounds  sterling, 
and  directed  that  the  funds  should  be  applied  to  "  the  instituting  and 
settling  a  professor  of  mathematics  and  experimental  philosophy  in 
Harvard  College."  To  the  i^ame  benefactor  Harvard  was  indebted 
for  the  establishment  of  the  professorship  of  divinity.  Down  to  the 
commencement  of  the  nineteenth  century  only  one  additional  professor 
was  appointed  in  the  undergraduate  department,  namely,  the  Hancock 
professor  of  Hebrew,  in  1765.  Hence,  it  follows  that  almost  all  regular 
instruction  was  given  by  tutors.  Previous  to  the  establishment  of  the 
Hollis  professorship  the  mathematical  instruction  was  entirely  in  the 
hands  of  tutors.  Since  almost  any  minister  was  considered  competent 
to  teach  mathematics,  and  since  tutors  held  their  place  sometimes  for 


24  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

only  one  year,  we  may  imagine  that  the  teaching  was  not  of  a  very  high 
order. 

The  first  appointment  to  the  Hollis  professorship  of  mathematics  and 
natural  philosophy  was  that  of  Isaac  Greenwood.  He  was  the  first  to 
occupy  a  collegiate  chair  of  mathematics  in  New  England,  but  not  the 
first  in  America,  as  is  sometimes  stated.  This  honor  beloDgs  to  a  pro- 
fessor at  William  and  Mary  College.  Greenwood  graduated  at  Harvard 
in  1721,  then  engaged  in  the  study  of  divinity,  visited  England,  and  be- 
gan to  preach  in  London  with  some  approbation,*  He  also  attended 
lectures  delivered  in  that  metropolis  on  experimental  philosophy  and 
mathematics.  In  1727  he  entered  upon  his  duties  at  Harvard.  "In 
scientific  attainments  Greenwood  seems  to  have  been  well  qualified  for 
his  professorship."  He  made  astronomical  contributions  to  the  Philo- 
sophical Transactions  of  1728,  and  published  in  1729  an  arithmetic. 
That  seems  to  have  been  the  earliest  arithmetic  from  the  pen  of  an 
American  author.  This  is  all  we  know  of  Greenwood  as  a  mathema- 
tician and  teacher.  Unfortunately  he  did  not  prove  himself  worthy  of 
Ms  place.  We  regret  to  say  that  the  earliest  professor  of  mathematics 
in  the  oldest  American  college  was  "  guilty  of  many  acts  of  gross  in- 
temperance, to  the  dishonor  of  God  and  the  great  hurt  and  reproach  of 
the  society."  His  intemperance  brought  about  his  removal  from  his 
chair  in  1738. 

On  the  dismissal  of  Greenwood,  Nathaniel  Prince,  who  had  been  tutor 
for  thirteen  years,  aspired  to  the  professorship.  He  was,  says  Elliot, 
superior  "  to  any  man  in  New  England  in  mathema.tics  and  natural 
philosophy."  But  his  habits  being  notoriously  irregular,  John  Win- 
throp  of  Boston,  was  appointed  in  his  stead.  Winthrop  graduated  at 
Harvard  in  1732,  and  was  only  twenty-six  years  old  when  he  was  chosen 
professor  of  mathematics  and  natural  philosophy.  He  filled  this  chair 
for  over  forty  years  (until  1779)  with  marked  ability.  In  mathematical 
science  he  came  to  be  regarded  by  many  the  first  in  America. 

If  we  could  turn  the  wheel  of  time  backward  through  one  hundred 
and  twenty  revolutions,  and  then  enter  the  leeture-room  of  Professor 
Winthrop  and  listen  to  his  instruction,  what  a  chapter  in  the  his- 
tory of  mathematical  teaching  would  be  uncovered!  But  as  it  is,  this 
history  is  hidden  from  us.  We  know  only  that  during  the  early  part 
of  his  career  as  professor,  "  and  probably  many  years  before,"  the  text- 
books were  the  following :  Ward's  Mathematics,  Gravesande's  Philos- 
ophy, and  Baclid's  Geometry  j  besides  this,  lectures  were  delivered  by 
the  professors  of  divinity  and  matheraatics.t 

From  this  we  see  that  some  time  between  the  years  1728  and  1738, 
Ward's  Mathematics  had  been  introduced,  and  Alsted's  old  Geometry 
had  given  place  to  the  still  older  but  ever  standard  work  of  Euclid. 
This  is  the  first  mention  of  Euclid  as  a  text-book  at  Harvard.    The  in- 

^  Quiucy's  History  of  Harvard  University,  Vol.  II,  p.  14. 
t  Peirce'a  History  of  Harvard,  p.  237. 


COLONIAL    TIMES.  25 

trod  action  of  G-ravesaiide's  Philosophy  is  another  indication  of  progress. 
Gravesaude  was  for  a  time  professor  of  mathematics  and  astronomy  at 
the  LTniversity  of  Leyden.  He  was  the  first  who  on  the  continent  of 
Europe  publicly  taught  the  philosophy  of  Fewton,  and  he  thus  con- 
tributed to  bring  about  a  revolution  in  the  physical  sciences.  By  the 
adoption  of  his  philosophy  as  a  text-book  at  Harvard  we  see  that  the 
teachings  of  Newton  had  at  last  secured  a  firm  footing  there.  Ward's 
Mathematics  continued  for  a  long  time  to  be  a  favorite  text-book.* 

It  is  probable  that  with  the  introduction  of  Ward's  Mathematics,  alge- 
bra began  to  be  studied  at  Harvard.  The  second  part  of  the  Young 
Mathematician's  Guide  consists  of  a  rudimentary  treatise  on  this  subject. 
It  is  possible,  then,  that  the  teaching  of  algebra  at  Cambridge  may  have 
begun  some  time  between  1726  and  1738.  But  I  have  found  no  direct 
evidence  to  show  that  algebra  actually  was  in  the  college  curriculum 
previous  to  1786. 

Since  Ward's  Mathematics  were  used,  to  our  knowledge,  not  only  at 
Harvard,  but  also  at  Yale,  Brown,  and  Dartmouth,  and  as  a  book  of 
reference  at  the  University  .of  Pennsylvania,  a  description  of  the  Young 
Mathematician's  Guide  may  not  be  out  of  place.t 

The  first  part  treats  of  arithmetic  (143  pages).  Though  very  deficient 
according  to-modern  notions,  the  presentation  of  this  subject  is  superior 
to  that  in  Dilworth's  School-master's  Assistant.    It  is  less  obscure. 

*  According  to  ex-Presidenfc  D.  Woolsey,  the  author  of  this  book  was  tlie  Ward 
who  had  been  "president  of  Trinity  College,  Cambridge,  and  bishop  of  Exeter." 
(Yale  College  ;  A  Sketch  of  its  History,  William  L,  Kingsley,  Vol.  II,  p.  499.)  Now, 
the  only  individnal  answering  to  this  description  is  Seth  Ward,  the  astronomer, 
whose  time  of  activity  preceded  the  epoch  of  Newton.  We  shall  show  that  the  book 
in  question  was  not  written  by  Seth  Ward,  but  by  John  Ward,  who  flourished  half 
a  century  later  than  Seth  Ward  and  whose  Young  Mathematician's  Guide  was  for  a 
long  time  a  popular  elementary  text-book  in  England.  Wherever  we  have  seen 
Ward's  book  mentioned  in  the  curricula  of  American  colleges  it  was  always  called 
"Ward's  Mathematics."  The  baptismal  name  of  the  author  was  never  given.  This 
shows  that  there  was  only  one  Ward  (either  Seth  or  John)  whose  mathematical  books 
were  known  and  used  in  our  colleges.  Now,  Benjamin  West,  professor  of  mathematics 
in  Brown  University  from  1786  to  1799,  published  in  the  first  volume  of  the  American 
Academy  of  Arts  and  Sciences  a  paper  "On  the  extraction  of  roots,"  in  which  he 
offers  improvements  on  "Ward's"  method.  Now,  I  have  seen  a  copy  of  Seth  Ward's 
Astronomia  Geometrica,  but  have  found  nothing  in  it  on  root  extraction.  One  would 
hardly  expect  to  find  anything  on  it  in  Seth's  "Trigonometry"  or  "Proportion." 
John  Ward,  on  the  other  hand,  treats  of  roots  in  his  "Guide,"  and  gives  a  "general 
method  of  extracting  roots  of  all  single  powers."  West  takes  two  examples  (two 
numbers,  one  of  14,  the  other  of  18  digits)  from  "Ward,"  and  shows  how  the  reauired 
roots  can  be  extracted  by  his  method.  But  both  these  examples  are  given  in  John 
Ward's  Young  Mathematician's  Guide.  This  evidence  in  favor  of  John  Ward's  book 
may  be  considered  conclusive.  Further  information  on  "Ward's  Mathematics"  will 
be  found  in  an  article  by  the  writer  in  the  Papers  of  the  Colorado  College  Scientific 
Society,  Vol.  I. 

tThe  copy  which  the  writer  hag  before  him  (Twelfth  edition,  London,  1771),  was 
kindly  lent  him  by  Dr.  Artemas  Martin,  of  the  U.  S.  Coast  Survey,  who  has  for  years 
been  making  a  collectiou  of  old  and  rare  books  on  mathematics. 


26  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

Like  all  books  of  that  time,  it  contains  rules,  but  no  reasoning.  What 
seems  strange  to  us  is  the  fact  that  subjects  of  no  value  to  the  begin- 
ner, such  as  arithmetical  and  geometrical  proportion  {i.  e.,  progression), 
alligation,  square  root,  cube  root,  biquadrate  root,  sursolid  root,  etc., 
are  given  almost  as  much  space  and  attention  as  common  and  decimal 
fractions. 

The  second  part  (140  pages)  is  devoted  to  algebra.  Ward  had  pub- 
lished a  small  book  on  algebra  in  1698,  but  that,  he  says,  was  only  "  a 
compendium  of  that  which  is  here  fally  handled  at  large."  Like  Har- 
riot, he  speaks  of  his  algebra  as  "Arithmetick  in  species."  This  name 
is  appropriate,  inasmuch  as  he  does  not  (at  least  at  the  beginning)  rec- 
ognize the  existence  of  negative  quantities,  but  speaks  of  the  minus  sign 
always  as  meaning  only  subtraction,  as  in  arithmetic.  A  little  further 
on,  however,  he  brings  in,  by  stealth,  "affirmative"  and  "negative" 
quantities.  The  knowledge  of  algebra  to  be  gotten  from  this  book  is 
exceedingly  meagre.  Factoring  is  not  touched  upon.  The  rule  of  signs 
in  multiplication  is  proved,  but  further  on  all  rules  are  given  without 
proof.  The  author  develops  a  rule  showing  how  binomials  can  be 
raised  "to  what  height  you  please  without  the  trouble  of  continued  in- 
volution." He  then  says :  "  I  proposed  this  method  of  raising  powers 
in  my  Compendium  of  Algebra,  p.  57,  as  wholly  new  (viz,  as  much  of  it 
as  was  then  useful),  having  then  (I  profess)  neither  seen  the  way  of 
doing  it,  nor  so  much  as  heard  of  its  being  done.  But  since  the  writing 
of  that  tract,  I  find  in  Doctor  Wallis's  History  of  Algebra,  pp.  319  and 
331,  that  the  learned  Sir  Isaac  l^ewton  had  discovered  it  long  before." 
The  subject  of  "  interest "  is  taught  in  the  book  algebraically,  by  the  use 
of  equations. 

Part  III  (78  pages)  treats  of  geometry.  In  point  of  precision  and 
scientific  rigor,  this  is  quite  inferior.  After  the  definitions  follow 
twenty  problems,  intended  for  the  excellent  purpose  of  exercising  the 
"  young  practitioner,"  and  bringing  "his  hand  to  the  right  manage- 
ment of  a  ruler  and  compass,  wherein  I  would  advise  him  to  be  very 
ready  and  exact."  Then  follows  a  collection  of  twenty-four  "  most  use- 
ful theorems  in  plane  geometry  demonstrated."  This  part  is  semi-em- 
pirical and  semi-demonstrative.  A  few  theorems  are  assumed  and  the 
rest  proved  by  means  of  these.  The  theorem,  "If  a  right  line  cut  two 
parallel  lines,  it  will  make  the  opposite  {i.  e.,  alternate  interior)  angles 
equal  to  each  other,"  is  proved  by  aid  of  the  theorem,  that  "  If  two 
lines  intersect  each  other,  the  opposite  angles  will  be  equal."  The  proof 
is  based  on  the  idea  that  "  parallel  lines  are,  as  it  were,  but  one  broad 
line,"  and  that  by  moving  one  parallel  toward  the  other,  the  figure  for 
the  former  theorem  reduces  to  that  of  the  latter.  The  next  chapter 
contains  the  algebraical  solution  of  twenty  geometrical  problems. 

Part  IV,  on  conic  sections  (36  pages),  gives  a  semi-empirical  treat- 
ment of  the  subject.  Starting  with  the  elefinition  of  a  cone,  it  shows 
how  the  three  sections  are  obtained  from,  it,  and  then  gives  some  of 
their  principal  properties. 


COLONIAL   TIMES.  27 

Part  V  (36  pages)  is  on  the  arithmetic  of  infinites.  Judging  from 
tbis  part  of  the  book,  its  author  knew  nothing  of  fluxions.  The  first 
edition  appeared  in  1707,  after  Newton  had  published  the  first  edition 
of  his  Priucipia,  in  1687,  but  his  Method  of  Fluxions  was  not  published 
till  1736,  though  written  in  1671.  Ward  employs  the  method  of  inte- 
gration bj  series  of  Oavalieri,  Roberval,  and  John  Wallis,  and,  thereby, 
finds  the  snperfleial  and  &olid  contents  of  solid  figures.  It  does  not  ap- 
pear that  this  pai^  of  the  book  was  ever  studied  in  American  colleges. 

Ward's  book  met  with  favor  in  England.  In  the  preface  to  the 
twelfth  edition  he  says :  ^'I  believe  I  may  truly  say  (without  vanity) 
this  treatise  hath  proved  a  very  helpful  guide  to  near  five  thousand  per- 
sons, *  *  *  and  not  only  so,  but  it  hath  been  very  well  received 
amongst  the  learned,  and  (I  have  been  often  told)  so  well  approved 
on  at  the  universities,  in  England,  Scotland,  and  Ireland,  that  it  is 
ordered  to  be  publicly  read  to  their  pupils." 

In  former  times  all  professors  of  mathematics  in  American  colleges 
gave  instruction,  not  merely  in  pure  mathematics,  but  also  in  natural 
philosophy  and  astronomy ;  and  it  appears  that  as  a  general  rule  these 
professors  took  more  real  interest  and  made  more  frequent  attempts  at 
original  research  in  the  fields  of  astronomy  and  natural  philosophy  than 
in  pure,  mathematics.  The  main  reason  for  this  lies  probably  in  the 
fact  that  the  study  of  pure  mathematics  met  with  no  appreciation  and 
encouragement.  Original  work  in  abstract  mathematics  would  have 
been  looked  upon  as  useless  speculations  of  idle  dreamers.  The  scien- 
tific activity  of  John  Winthrop  was  directed  principally  to  astronomy. 
His  reputation  abroad  as  a  scientist  Was  due  to  his  work  in  that  line. 
In  1740  he  made  observations  on  the  transit  of  Mercury,  which  were 
printed  in  the  Transactions  of  the  Eoyal  Society.  In  1761  there  was  a 
transit  of  Venus  over  the  sun's  disk,  and  as  Newfoundland  was  the  most 
western  part  of  the  earth  where  the  end  of  the  transit  could  be  ob- 
served, the  "province"  sloop  was  fitted  out  at  the  public  expense  to  con- 
vey Winthrop  and  party  to  the  place  of  observation.*  He  took  with 
him  two  pupils  who  had  made  progress  in  mathematical  studies.  One 
of  these,  Samuel  Williams,  became  later  his  successor  at  Harvard.  In 
1769  Winthrop  had  another  chance  for  observing  the  transit  of  Venus, 
at  Cambridge.  "As  it  was  the  last  opportunity  that  generation  could 
be  favored  with,  he  was  desirous  to  arrest  the  attention  of  the  peo- 
ple. He  read  two  lectures  upon  the  subject  in  the  college  chapel, 
which  the  students  requested  him  to  publish.  The  professor  put  this 
motto  upon  the  title  page:  Agite,  mortales!  et  oculos  in  spectaculum  ver- 
tite,  quod  Imcusqiie  spectaverunt  perpaucissimi ;  spectaturi  iterum  sunt 
nulli."  (Come,  mortals!  and  turn  your  eyes  upon  a  sight  which,  to  this 
day,  but  few  have  seen,  and  which  not  one  of  us  will  ever  see  again.) 
The  transit  of  1769  was  also  observed  in  Philadelphia  by  David  Eit- 
tenhouse,  and  in  Providence  by  Benjamin  West.    These  observations 

*  "  Jolm  Winthrop,"  in  the  Biographical  Dictionary  by  John  Eliot,  1809. 


28  TEACHING   AKD   HISTORY    OF   MATHEMATICS. 

were  an  important  a-id  in  determining  the  sun's  parallax.  Most  grati- 
fying to  us  is  the  interest  in  astronomical  pursuits  manifested  in  those 
early  times.  Expeditions  fitted  out  at  public  expense,  and  private  mu- 
nificence in  the  purchase  of  suitable  instruments,  bear  honorable  testi- 
mony to  the  enlightened  seal  which  animated  the  friends  of  science. 

In  1767  John  Winthrop  wrote  his  Gogita  de  Gometis,  which  he  dedi- 
cated to  the  Royal  Society,  of  which  he  had  been  elected  a  member. 
This  was  reprinted  in  Loudon  the  next  year,  and  gave  him  an  extensive 
literary  rei)utation. 

In  1764  a  calamity  befell  Harvard  College.  The  library  and  philo- 
sophical apparatus — the  collections  of  over  a  century — were  destroyed 
by  fire.  Among  the  books  recorded  as  having  been  lost  are  the  follow- 
ing :  "  The  Transactions  of  the  Eoyal  Society,  Academy  of  Sciences  in 
France,  Acta  Eruditorum,  Miscellanea  Curiosa,  the  works  of  Boyle 
and  Newton,  with  a  great  variety  of  other  mathematical  and  philo- 
sophical treatises."*  It  is  seen  from  this  that,  before  the  fire,  books  of 
reference  in  higher  mathematics  had  not  been  entirely  wanting. 

John  "Winthrop  died  in  1779,  and  the  robe  of  the  departing  prophet 
fell  upon  his  former  disciple,  the  Eev.  Samuel  Williams.  Williams  filled 
the  mathematical  chair  for  eight  years.  Having  inherited  from  his  mas- 
ter a  love  of  astronomy,  he  frequently  published  observations  and  no- 
tices of  extraordinary  natural  phenomena  in  the  memoirs  of  the  Ameri- 
can Academy  of  Arts  and  Sciences.  He  occupied  the  mathematical  chair 
at  Harvard  until  17S8.  Then  he  lectured  at  the  University  of  Vermont 
on  astronomy  and  natural  philosophy  for  two  years,  and  was  subse- 
quently minister  at  Eutland  and  Burlington,  Vermont. 

YALE  COLLEGE. 

Yale,  the  second  oldest  New  England  college,  was  founded  in  1701, 
or  sixty-three  years  after  the  opening  of  Harvard.  During  the  first 
fifteen  years  it  maintained  a  sort  of  nomadic  existence.  Previous  to  1816 
instruction  seems  to  have  been  given  partly  at  Saybrook  and  partly  at 
Killingworth  and  Milford.  Its  course  of  instruction  was  then  very 
limited.  The  mathematical  teaching  during  the  first  years  of  its  exist- 
ence was  even  more  scanty  than  in  the  early  years  at  Harvard.  Benja- 
min Lord,  a  Yale  graduate  of  1714,  wrote  in  1779  as  follows  in  reply  to 
inquiries  by  President  Stiles :  "  As  for  mathematics,  we  recited  and 
studied  but  little  more  than  the  rudiments  of  it,  some  of  the  plainest 
things  in  it.  Our  advantages  in  that  way  were  too  low  for  any  to  rise 
high  in  any  brancn  of  literature."  t  Doctor  Johnson,  of  the  same  class, 
says :  "  Common  Arithmetick  and  a  little  surveying  were  the  ne  plus 
ultra  of  mathematical  acquirements."  It  appears  from  this  that  sur- 
'Dei/lng  was  taken  at  Yale,  instead  of  the  geometry  which  formed  part 

*Vide  Quiucy's  History  of  Harvard  University,  Vol.  II,  p.  481. 
tYale  Biographiea  aud  Aunals,  1701-45,  by  Frauklin  Bowditch  Dexter,  pp.  115 
and  116. 


COLONIAL   TIMES.  29 

of  tlie  course  at  Harvard.  In  a  new  and  only  partially  settled  country 
some  knowledge  of  surveying  was  a  great  desideratum.  Bat  the  study 
of  surveying  without  a  preliminary  course  in  geometry  and  trigonometry 
is  truly  characteristic  of  the  purely  practical  tendencies  of  the  times. 
Men  took  eager  interest  in  the  applications  of  science,  but  cared  nothing 
for  science  itself.  The  little  mathematics  studied  was  evidently  not 
pursued  for  its  own  sake,  nor  for  the  mental  discipline  which  it  afforded, 
but  simply  for  the  pecuniary  profit  which  it  would  afterwards  bring. 

As  at  Harvard,  so  at  Yale,  the  mathematics  were  studied,  at  that 
time,  during  the  last  year  of  the  college  course  and  after  the  study  of 
physics  had  been  completed.*  Duringthe  next  six  or  seven  years,  the 
course  at  Yale  was  extended  somewhat.  In  1720  it  was  identical  with 
the  Harvard  course  of  1726.  In  1719,  when  Jonathan  Edwards  was  a 
member  of  the  Junior  class  at  New  Haven,  he  wrote  as  follows  to  his 
father:  "  I  have  enquired  of  Mr.  Cutter,  what  books  we  shall  have  need 
of  the  next  year.  He  answered  he  would  have  me  get  against  that  time, 
Alsteds'  Geometry  and  Gassendi's  Astronomy."! 

At  this  time  progress  was  also  made  in  the  teaching  of  physics.  The 
earliest  guide  in  this  study  were  the  manuscript  lectures  by  Sector 
Pierson,  which  were  a  repetition  of  lectures  he  had  heard  while  a 
student  at  Harvard  College.  They  were  metaphysical  rather  than 
mathematical,  "recognizing  the  Copernican  theory,  but  knowing  nothing 
of  Kepler  and  Galileo,  and  much  less  of  ]S"ewtou."| 

During  the  first  seventeen  years  at  Yale  the  doctrines  of  the  school- 
men in  logic,  metaphysics,  and  ethics  still  held  sway.  Descartes, 
Boyle,  Locke,  Bacon,  and  Newton  were  rega,rded  as  innovators  from 
whom  no  good  could  be  expected.  It  is  pleasing  to  think  that  the  in. 
troduction  of  Newtonian  ideas  and  the  rise  of  mathematical  studies  at 
Yale  was  partly  due  to  an  act  of  charity  by  the  great  Sir  Isaac  Newton 
himself.  In  the  year  1715  a  collection  of  books  made  in  England  by 
Mr.  Drummer,  the  agent  of  the  colony,  amounting  to  about  eight 
hundred  volumes,  was  sent  over  to  the  college.  The  collection  con- 
sisted of  donations  by  well-spirited  gentlemen  in  Britain.  "  Sir  Isaac 
Newton  gives  the  second  edition  of  his  Principia  (which  appeared  in 
1713)";  "  Doctor  Halley  sends  his  edition  of  Apollonius."§  But  these 
and  many  other  donations  would  have  been  barren  of  results  had  there 
not  been  young  men  of  talent  a^nd  energy  to  master  the  contents  of 
these  precious  volumes.  Such  a  man  was  Samuel  Johnson.  He  gradu- 
ated in  1714  and  was  appointed  tutor  a  few  years  later.  Drummer's 
collection  furnished  him  with  a  "  feast  of  fat  things."  To  use  his  own 
words:  *'He  seemed  to  himself  like  a  person  suddenly  emerging  out  of 
the  glimmer  of  twilight  into  the  full  sunshine  of  open  day."    He  and 

*  Yale  College ;  a  sketch  of  its  history,  William  L.  Kingsley,  Vol.  II,  p.  496. 

t  Edwards'  Works,  Vol.  I,  p.  30. 

t  Ex-President  D.  Woolsey,  in  Yale  Book,  Vol.  II,  p.  499. 

$Yale  Biographies  aud  Annals,  1701-45,  by  Franklin  Bowditch  Dexter,  p.  141. 


30  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

Mr.  Brown,  another  young  tutor,  exerted  themselves  to  the  utmost  for 
the  improvement  of  the  students  under  their  charge.  Imbued  with  the 
grand  ideas  of  Newton,  they  extended  the  mathematical  course  for  the 
understanding  of  the  Newtonian  system,  and  then  taught  this  system 
in  place  of  the  older.  There  was  at  that  time  much  contention  as  to 
the  place  where  the  college  should  be  permanently  located.  This  was 
a  fortunate  circumstance  for  the  young  tutors,  since  these  troubles 
without  withdrew  public  attention  from  the  innovations  witbin.*  In 
1722  Johnson  and  Brown  resigned  their  tutorships  and  sailed  for  Eng- 
land to  receive  ordination  from  an  English  bishop.  Johnson  became 
later  president  of  King's  (now  Columbia)  College  in  New  York. 

Soon  after  this  the  Physics  of  Eohault  was  introduced  at  Yale  as  a 
text-book.  Eohault  (1620-75)  was  a  French  philosopher  and  an  im- 
plicit follower  of  the  Cairtesian  theory.  The  edition  used  was  that  by 
the  celebrated  Samuel  Clarke,  who  had  taken  the  rugged  Latin  version 
of  the  treatise  of  Eohault  (then  used  as  a  text-book  at  the  University 
of  Cambridge,  England),  and  published  it  in  better  Latin,  together 
with  numerous  critical  notes,  which  he  had  added  with  a  view  of 
bringing  the  Cartesian  system  into  disrepute  by  exposing  its  fallacies. 
This  disguised  Newtonian  treatise  maintained  its  place  at  Yale  until 
1743,  when  it  was  superseded  by  the  work  of  Gravesande. 

During  President  Clap's  time,  Martin's  Philosophy,  in  three  volumes, 
was  the  text-book  in  this  science ;  when  this  work  came  to  be  out  of 
print.  President  Stiles  procured  Enfield's  Philosophy,  which  was  the 
first  introduction  into  j^merican  colleges  of  that  now  obsolete  work. 

It  is  worthy  of  remark  that,  in  1749,  Benjamin  Eranklin  presented  to 
the  college  an  electric  machine,  and  that,  a  few  years  later,  Ezra  Stiles, 
then  tutor  at  the  college,  began  to  make  experiments  with  it.  These 
are  supposed  to  have  been  the  earliest  of  the  kind  made  in  New  England. 

It  appears  that  in  1733,  Euclid  was  being  used  as  a  text-book  in 
geometry.  The  earliest  mention  of  Euclid  at  Harvard  is  in  1737.  In 
1733,  Dr.  John  Hubbard  of  New  Haven,  who  had  received  the  honorary 
degree  of  master  of  arts  three  years  previously,  testified  his  gratitude 
by  writing  a  panegyric,  "The  Benefactors  of  Yale  College." 

He  introduced  a  recent  gift  of  mathematical  books  by  Josei)h  Thomp- 
son, of  London,  with  the  following  stanza: 

"  The  Mathematicks  too  our  tlio'ts  employ, 
Which  nobly  elevate  the  Student's  joy: 
The  little  Euclids  round  the  table  set 
And  at  their  rigid  demonstrations  sweat."  t 

This  same  Joseph  Thompson  donated  to  the  college  also  "a  complete 
set  of  surveying  instruments,  valued  at  £21."  "  A  reflecting  telescope, 
a  microscope,  a  barometer,  and  other  mathematicalinstrumeuts — valued 
at  £37,  were  bought  by  a  subscription  from  the  trustees  and  others."| 

*  Barnard's  Journal,  Vol.  XXVH,  1877;  Article:    "Samuel  Johnson." 
t  Yale  Biographies  and  Annals^  1701-45,  by  F.  B.  Dexter,  p.  473. 
t  Ibid.,  p.  521.  f 


COLONIAL   TIMES.  31 

In  1742,  elementary  mathematics  came  to  be  removed  from  its  august 
position  in  the  curriculum  as  a  senior  study,  and  to  be  assigned  an 
humbler  but  more  befitting  place  nearer  the  beginning  of  the  course. 
In  1742  the  rector  of  the  college  advised  the  students  to  pursue  a  regular 
course  of  academic  studies  in  the  following  order :  "  In  the  first  year  to 
study  principally  the  tongues,  arithmetic,  and  algebra;  the  second, 
logic,  rhetoric,  and  geometry;  the  third,  mathematics,  and  natural 
philosophy ;  and  the  fourth,  ethics  and  divinity."* 

That  these  changes  were  not  made  earlier  than  1742  is  evident  from 
a  passage  in  the  memoir  of  Samuel  Hopkins,  who  graduated  in  1741, 
stating  that  then  "  metaphysics  and  mathematics  found  their  place  in 
in  the  fourth  year,  being  in  their  turn  the  subject  of  study  and  recita- 
tion for  the  first  four  days  of  every  week."t 

At  what  time  this  dethronement  of  elementary  mathematics  as  a 
senior  study  took  place  at  Harvard,  we  are  not  able  to  state.  It  will 
be  noticed  that,  at  Yale,  mathematics  and  natural  philosophy  had  at 
this  period  exchanged  places,  the  former  now  preceding  the  latter. 
From  the  above  it  is  also  evident  that  algebra  was  studied  at  Yale  in 
1742.  The  earliest  mention  of  algebra  at  Cambridge  is  in  1786,  though 
it  doubtless  began  to  be  taught  there  much  earlier.  What  branch  of 
mathematics  constituted  the  study  for  the  third  or  Junior  year  remains 
a  matter  of  conjecture.  The  "mathematics"  spoken  of  in  the  extract 
probably  referred  to  trigonometry,  possibly  together  with  some  other 
branches. 

A  strong  impetus  to  the  study  of  mathematics  at  Yale  was  given 
during  President  Clap's  administration.  Thomas  Clap  graduated  at 
Harvard  in  1722.  Doctor  Stiles,  his  successor  in  the  presidency  at 
Yale,  says  that  Clap  studied  the  higher  branches  of  mathematics,  and 
was  one  of  the  first  philosophers  America  has  produced,  "  that  he  was 
equalled  by  no  man,  except  the  most  learned  Professor  Winthrop."  In 
his  history  of  Yale,  written  in  1766,  the  year  of  his  resignation.  President 
Clap  gives  the  following  account  of  the  studies  pursued  by  students  at 
the  college : 

*'In  the  first  year  they  learn  Hebrew,  and  principally  pursue  the 
study  of  the  languages,  and  make  a  beginning  in  logic  and  some  parts 
of  the  mathematics.  In  the  second  year  they  study  the  languages,  but 
principally  recite  logic,  rhetoric,  oratory,  geography,  and  natural  phi- 
losophy ;  and  some  of  them  make  good  proficiency  in  trigonometry  and 
algebra.  In  the  third  year  they  will  pursue  the  study  of  natural  phi- 
losophy and  most  branches  of  mathematics.  Many  of  them  well  under- 
stand surveying,  navigation,  and  the  calculation  of  eclipses ;  and  some 
of  them  are  considerable  proficients  in  conic  sections  and  fluxions.  In 
the  fourth  year  they  principally  study  and  recite  metaphysics,  ethics, 
and  divinity."! 

*Yale  Biographies  and  Annals,  1701-45,  p.  724. 

tNew  Englander,  August,  1852,  p.  452:  Professor  Park's  Memoir  of  Hopkins. 

%  Yale  College ;  a  Sketch  of  its  History,  by  Win.  L.  Kiugsley,  Vol.  II,  pp.  497  and  498. 


32  TEACHING    AND    HISTOSY    OF   MATHEMATICS. 

The  mathematical  coarse  in  the  above  curriculum  is  indeed  one  that 
Tale  had  reason  to  be  proud  of.  It  shows  that  not  only  algebra  and 
geometry,  but  also  trigonometry,  and  even  conic  sections  and  fluxions^ 
were  studied  at  Yale  previous  to  the  year  176G.  This  is  the  earliest 
distinct  mention  of  conic  sections  aud  flusions  as  college  studies  in 
America. 

Mathematics  seem  to  have  come  to  occupy  some  of  the  time  which 
was  given  at  first  to  logic.  President  Olap  does  not  enumerate  the 
text-books  employed,  but  his  successor,  Doctor  Stiles,  in  his  (liary  for 
November  9, 1779,  mentions  a  list  of  books  recited  in  the  several  classes 
at  his  accession  to  the  presidency,  in  1777.  The  mathematical  books 
are,  for  the  Freshman  class, Ward's  Arithmetic ;  Sophomore  class,  Ham- 
mond's Algebra,  Ward's  Geometry  (Saturday),  Ward's  Mathematics  ; 
Junior  class,  Ward's  Trigonometry,  Atkinson  and  Wilson's  Trigonom- 
etry. 

On  comparing  this  mathematical  course  with  that  given  by  President 
Clap  eleven  years  previous  we  observe  some  changes.  The  study  of 
conic  sections  and  fluxions  had  been  apparently  discontinued.  Tliis 
waning  of  ma^thematical  enthusiasm  was  probably  due  to  the  departure 
of  President  Clap,  and  also  to  the  political  disturbances  and  confusions 
of  the  times.  It  would  seem  that  during  Clap's  administration  not  all 
the  students  took  higher  mathematics,  but  only  those  who  were  partic- 
ularly fond  of  them.  Clap  says,  "■  Many  of  tliem  well  understand  sur- 
veying, navigation,  and  the  calculation  of  eclipses;  and  some  of  them 
are  considerable  proficients  in  the  conic  sections  and  fluxions." 

That  optional  studies  were  then  pursued  occasionally  is  evident  from 
a  statement  by  President  Stiles  that  he  began  instructing  a  class  in  He- 
brew and  Oriental  languages, *which  he  "selected  out  of  all  other 
classes,  as  they  voluntarily  offered  themselves."  The  extent  to  which 
each  of  these  branches  was  studied  may  probably  be  correctly  inferred 
from  the  contents  of  Ward's  Young  Mathematician's  Guide.  This  con- 
sists of  five  parts :  arithmetic,  algebra,  geometry,  conic  sections,  and 
arithmetic  of  infinites.  Students  that  were  mathematically  inclined 
went  through  the  entire  work  it  would  seem,  excepting  the  algebra, 
which  was  studied  from  Hammond's  book. 

The  year  1770  is  memorable  for  the  creation  of  the  chair  of  "  mathe- 
matics and  natural  philosophy  "  at  Yale.  This  was  done  apparently  to 
fill  the  gap  caused  by  the  departure  of  President  Clap,  who  was  uncom- 
monly skilled  in  those  sciences.  The  first  occupant  of  this  chair  was 
IsTehemiah  Strong,  who  kept  it  eleven  years.  He  belonged  to  the  class 
of  1755  at  Yale,  and  was  tutor  there  from  1757  to  '60.  Before  entering 
upon  the  duties  of  his  chair,  he  had  been  pastor.  After  his  resigna- 
tion of  his  chair,  he  entered  upon  the  study  and  practice  of  law.  He 
published  an  "Astronomy  Improved"  (New  Haven,  1784).  President 
T.  Dwight  speaks  of  him  as  "a  man  of  vigorous  undeistaudiug." 


COLONIAL    TIMES.  33 

WILLIAM  AND  MARY  COLLEGE. 

William  and  Mary  is  next  to  Harvard  the  oldest  of  American  col- 
leges. From  1688,  the  year  of  its  organization  at  Williamsburg,  Va., 
until  the  inauguration  of  the  University  of  Yirginia,  it  was  the  leading 
educational  institution  in  the  South.  Owing  to  the  repeated  destruc- 
tion by  fire  of  the  college  buildings  and  records,  not  even  the  succes- 
sion of  the  professors  has  been  preserved.  The  early  courses  were  in 
all  probability  much  the  same  as  the  contemporaneous  courses  at  Har- 
vard. According  to  Campbell,  5  professorships  were  provided  for  by 
the  charter,  namely,  those  of  Greek  and  Latin,  mathematics,  moral 
philosophy,  and  two  of  divinity.  In  speaking  of  the  early  course  of 
study,  Howison  says  that  it  embraced  also  a  "  natural  philosophy 
which  was  just  beginning  to  believe  that  the  earth  revolved  round  the 
sun,  rather  than  the*  sun  round  the  earth." 

The  earliest  mathematical  professor  at  William  and  Mary  whose  name 
has  come  down  to  us,  was  Rev.  Hugh  Jones.  The  college  had  a  'pro- 
fessorship of  mathematics  from  its  very  beginning,  and  at  a  date  when 
mathematical  teaching  at  Harvard  was  still  in  the  hands  only  of  tutors. 
The  names  of  the  predecessor  or  predecessors  of  Hugh  Jones  are  not 
known.  He  is  the  earliest pro/essor  of  mathematics  in  America  whose 
name  has  been  handed  down  to  us.  He  was  an  Englishman  of  univer- 
sity education ;  came  to  Maryland  in  1696 ;  was  for  a  time  pastor  of 
a  church;  and  then  was  appointed  to  the  chair  of  mathematics  at 
William  and  Mary.  He  was  a  man  of  broad,  scholarly  attainments,  and 
endeared  himself  to  the  student  of  history  quite  as  much  as  to  the 
mathematician,  by  writing  his  invaluable  book  on  The  Present  State  of 
Virginia  (1724).  Says  Dr.  Herbert  B.  Adams :  "  His  monograph  is 
acknowledged  to  be  one  of  the  best  sources  of  information  respecting 
Virginia  in  the  early  part  of  the  eighteenth  century."  The  following 
quotations  from  it  (p.  44)  may  be  of  interest :  "  They  (the  Virginians) 
are  more  inclinable  to  read  men  by  business  and  conversation  than  to 
dive  into  books,  and  are  for  the  most  part  only  desirous  of  learning 
what  is  absolutely  necessary  in  the  shortest  and  best  method." 

"  Having  this  knowledge  of  their  capacities  and  inclination  from  suf- 
ficient experience,  I  have  composed  on  purpose  some  short  treatises 
adapted  with  my  best  judgment  to  a  course  of  education  for  the  gentle- 
men of  the  plantations,  consisting  in  a  short  Bnglish  Grammar,  an  Ac- 
cidenceofChristia7iity,  an  Accidence  to  the  Matliematick  in  all  its  parts 
and  applications.  Algebra,  Geometry,  Surveying  of  Land,  and  Naviga- 
tion.^'' 

I 

"  These  are  the  most  useful  branches  of  learning  for  them,  and  such 
as  they  wiliingly  and  readily  master,  if  taught  in  a  plain  and  short 
method,  truly  applicable  to  t\iQiY  genius;  which  I  have  endeavored  to 
do,  for  the  use  of  tJiem  and  all  others  of  their  temper  and  parts." 

We  are  not  to  understand  by  the  above  that  Ms  "Accidence  to  the 
881— No.  3 3 


34  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

Matheiuatick "  aud  thfe  other  books  mentioned  were  actually  printed ; 
they  existed  only  in  manuscript  copies.  From  the  above  it  appears  that 
about  1724  the  mathematical  course  at  William  aud  Mary  was  quite 
equal  to  that  in  either  of  the  two  New  England  colleges.  We  must,  of 
course,  guard  ourselves  against  the  impression  that  full  and  exhaustive 
courses  were  given  iu  algebra,  geometry,  surveying,  and  navigation. 
As  is  pointed  out  by  the  author  himself,  the  merest  rudiments  only 
were  imparted. 

Eevereud  Jones  was  succeeded  by  Alexander  Irvine,  and  he  in  turn 
by  Joshua  Fry.  Fry  was  educated  at  Oxford,  and,  after  coming  to  this 
country,  was  made  master  of  the  grammar  school  connected  with  Will- 
iam and  Mary,  and  later,  professor  of  mathematics  in  the  college.  In 
company  with  Peter  Jelferson,  the  father  of  Thomas  Jefferson,  he  made 
a  map  of  Virginia.  He  also  served  on  a  commission  appointed  to  deter- 
mine the  Virginia  and  North  Carolina  boundary  "line.  He  was  suc- 
ceeded in  1758  by  William  Small. 

A  few  years  before  the  outbreak  of  the  Eevolutionary  War  William 
and  Mary  College  had  among  her  students  several  who  afterwards  rose 
to  prominence ;  she  had  four  who  became  signers  of  the  Declaration  of 
Independence,  and  also  the  illustrious  Thomas  Jeiferson,  who  became 
the  author  of  this  great  document.  At  William  and  Mary,  Jefferson 
was  a  passionate  student  of  mathematics.  The  college  long  exercised 
the  duties  of  the  office  of  surveyor-general  of  the  Colony  of-  Virginia. 
Thomas  Jefferson's  father  was  a  practical  surveyor,  who  had  been 
chosen  in  1747  with  Joshua  Fry,  then  professor  of  mathematics  at  Will- 
am  and  Mary,  to  continue  the  boundary  line  between  Virginia  and 
North  Carolina. 

When  Thomas  Jefferson,  at  the  age  of  seventeen,  entered  the  Junior 
class,  he  came  into  intimate  relation  with  Dr.  William  Small,  of  Scotland, 
who  was  then  the  professor  of  mathematics.  As  an  instructor  he  had 
the  happy  gift  of  making  the  road  of  knowledge  both  easy  and  profit- 
able. In  his  Autobiography  Jefferson  says :  "  It  was  my  great  good 
fortune,  and  what  probably  fixed  the  destinies  of  my  life,  that  Dr.  Will- 
iam Small,  of  Scotland,  was  then  professor  of  mathematics,"  a  man  pro- 
found  in  most  of  the  useful  branches  of  science,  with  a  happy  talent  of 
communication,  correct  and  gentlemanly  manners,  and  an  enlarged  and 
liberal  mind.  He,  most  happily  for  me,  became  soon  attached  to  me, 
and  made  me  his  daily  coiupanion  when  not  engaged  in  the  school ;  and 
from  his  conversation  I  got  my  first  views  of  the  expansion  of  science, 
and  of  the  system  of  things  in  which  we  are  placed." 

In  1773  Thomas  Jefferson  was  appointed  surveyor  of  the  county  of 
Albemarle.  But  the  college  of  Williamsburg  left  its  stamp  upon  Jef- 
ferson, not  merely  as  a  qualified  surveyor,  but  also  as  a  statesman,  phi- 
losopher, economist,  and  educator.  We  dwell  with  special  interest  upon 
his  association  at  college  with  Dr.  Small,  because  in  later  years,  when 
filling  the  office  of  President  of  the  United  States,  we  shall  marvel  at 
the  rich  fruits  his  early  association  with  a  lover  of  exact  science  brought 


COLONIAL   TIMES.  35 

forth.  It  was  during  Jefferson's  administration  that  a  systematic  plan 
of  conducting  the  Government  surveys  of  the  great  Korth-West  Terri- 
tory was  initiated ;  it  was  during  his  administration  that  the  great  work 
of  the  U.  S.  Coast  Survey  was  first  inaugurated.  He  took  also  great 
interest  in  the  enlargement  of  the  U.  S.  Military  Academy.  In  these 
great  movements  the  personal  interest  and  enlightened  zeal  of  Jefferson 
himself  were  the  primary  motive  power.  His  biographers  tell  us  that 
he  was  the  first  discoverer  of  the  formula  for  constructing  the  mould- 
board  of  a  plow  on  mathematical  principles.  He  wrote  to  Jonathan 
Williams  on  this  subject,  July  3,  1796  :  "  I  have  a  little  matter  to  com- 
municate, and  will  do  it  ere  long.  It  is  the  form  of  a  mould  board  of 
least  resistance.  I  had  some  years  ago  conceived  the  principles  of  it,  and 
1  explained  them  to  Mr.  Eittenhouse."  We  quote  the  following  to  show 
that  even  in  his  old  age  he  still  loved  the  favorite  study  of  his  youth. 
Said  he  in  a  letter  to  Col.  William  Duane,  dated  October,  1812,  "  When 
I  was  young,  mathematics  was  the  passion  of  my  life.  The  same  pas- 
sion has  returned  upon  me,  but  with  unequal  powers.  Processes  which 
I  then  read  off  with  the  facility  of  common  discourse,  now  cost  me  labor 
and  time,  and  slow  investigation."  Of  interest  are  also  certain  pas- 
sages in  a  course  of  legal  study  which  he  drew  up  for  a  young  friend : 
"  Before  you  enter  on  the  study  of  law  a  sufficient  groundwork  must 
be  laid.  *  *  *  Mathematics  and  natural  philosophy  are  so  useful 
in  the  most  familiar  occurrences  of  life  and  are  so  peculiarly  engag- 
ing and  delightful  as  would  induce  every  one  to  wish  an  acquaint- 
ance with  them.  Besides  this,  the  faculties  of  the  mind,  like  the  mem- 
bers of  the  body,  are  strengthened  and  improved  by  exercise.  Mathe- 
matical reasoning  and  deductions  are,  therefore,  a  fine  preparation  for 
investigating  the  abstruse  speculations  of  the  law."  Among  the  books 
in  mathematics  recommended  by  Jefferson  to  his  young  friend  are, 
Bezout's  Cours  de  Mathematique — the  best  for  a  student  ever  published  j 
Montucla,  or  Bossut,  Histoire  des  Ilathmatiques ;  Astronomy — Fergu- 
son, and  Le  Monnier  or  De  Lalande. 

It  should  not  be  left  unmentioned  here  that  George  Washington  once 
applied  to  the  College  of  William  and  Mary  for  an  elective  course  in 
land  surveying,  and  that  he  received  his  first  commission  as  county 
surveyor  from  the  faculty  of  the  college.  In  this  connection  we  can  not 
refrain  quoting  a  passage  from  the  excellent  monograph  by  Dr.  Herbert 
B.  Adams  on  the  College  of  William  and  Mary.*  "  It  is  interesting," 
says  he,  "to  trace  the  evolution  of  men  as  well  as  of  institutions.  It  is 
generally  known  that  Washington  began  his  public  life  as  a  county 
surveyor,  but,  in  all  probability,  few  persons  have  ever  thought  of  his 
service  in  that  office  as  the  historical  and  economic  germ  of  his  political 
greatness.  Most  people  regard  this  early  work  as  a  passing  incident 
in  his  career,  and  not  as  a  determining  cause,  and  yet  it  is  possible  to 
show  that  Washington's  entire  public  life  was  but  the  natural  out- 

*  Circular  of  Information  of  the  Bureau  of  Education,  No.  1,  1887,  p.  30. 


36  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

growth  of  that  original  appointment  given  him  in  1749,  at  the  age  of 
seventeen,  by  the  College  of  William  and  Mary.  That  appointment,  in 
the  colonial  days  of  Virginia,  was  the  equivalent  of  a  degree  in  civil 
engineering,  and  it  is  interesting  to  observe  what  a  peculiar  bias  it 
gave  to  Washington's  life  before  and  after  the  Eevolution." 

Professor  Small's  successors  in  the  mathematical  chair  at  William 
and  Mary  were  Eev.  Thomas  Gwatkin,  George  Blackburn,  Ferdinand 
S.  Campbell,  Robert  Saunders,  Benjam'n  S.  Ewell,  and  Thomas  T.  L, 
Snead. 

UNIVERSITY  OF  PENNSTLYANIA. 

The  University  of  Pennsylvania  was  chartered  in  1755,  and  was 
known  before  the  Revolution  as  the  College,  Academy,  and  Charitable 
School  of  Philadelphia.  The  celebrated  Dr.  William  Smith,  D.  D.,  was 
the  first  provost.  He  was  a  man  of  great  learning  and  superior  execu- 
tive ability.  Under  his  administration,  previous  to  the  outbreak  of  the 
Revolution,  the  college  made  marvellous  progress.  The  teachers  were 
men  of  well-established  reputation  throughout  the  colonies.  Dr. 
Smith,  who  was  very  fond  of  mathematical  studies,  gave  lectures  on 
mathematics,  natural  x^hilosophy,  astronomy,  and  rhetoric.  In  1769  he 
appears  as  one  of  the  founders  of  the  American  Philosophical  So- 
ciety. The  first  volume  of  the  transactions  of  that  society  contains  ac- 
curate observations  by  Eittenhouse  and  himself  of  the  transits  of 
Venus  and  Mercury.  Associated  with  him  at  the  college  as  professor 
of  mathematics,  from  1760  to  1763,  was  Hugh  Williams.  He  was  a 
graduate  of  the  institution,  and  a  minister.  Afterward  he  studied 
medicine  abroad  and  then  practiced  in  Philadelphia.  He  took  great  in. 
terest  in  astronomy,  and  observed  the  transit  of  Yenus  and  Mercury 
for  the  Philosophical  Society. 

Theophilus  Grew  is  also  mentioned  as  a  mathematical  instructor. 
Eev.  Ebenezer  Kinnersley,  Franklin's  assistant  in  his  electrical  experi- 
ments, gave  instruction  in  physics.  *'  In  this  institution,"  says  Dr. 
Smith,  "  there  is  a  good  apparatus  for  experiments  in  natural  philoso- 
j)hy,  done  in  England  by  the  best  hands  and  brought  over  from  thence 
in  different  parcels.  There  is  also  in  the  experiment  room  an  electrical 
apparatus,  chieJiy  the  invention  Of  one  of  the  professors,  Mr.  Kinners- 
ley, and  perhaps  the  completest  of  the  kind  now  in  the  world."  The 
courses  of  study  mapped  out  by  Dr.  Smith  are  preserved  in  his 
works.*  According  to  this,  the  mathematical  and  physical  instruction 
during  the  three  years  at  college  was  as  follows  (in  1758) : 

First  year. — Common  and  decimal  arithmetic  reviewed,  including  fractions  and  the 

extraction  of  roots ;  algebra  through  simple  and  quadratic  equations,  -and  log- 

arithmical  arithmetic ;  first  six  hooka  of  Euclid. 
Second  year. — Plane  and  spherical  trigonometry;   surveying,  dialing,  navigation  ; 

eleventh  and  twelfth  books  of  Euclid;  conic  sections ;  fluxions;  architecture, 

with  fortification ;  physics. 
Third  year. — Light  and  color,  optics,  perspective,  astronomy. 

*  WiUiam  Smith's  Works,  1803,  p.  238. 


COLONIAL    TIMES.  37 

There  is  given,  in  addition  to  this,  the  following  list  of  ^'  books  recom- 
mended for  improving  the  youth  in  the  various  branches." 

First  2/ear.— Barrow's  Lectures,  Pardie's  Geometry,  Maclaurin's  Algebra,  Ward's 

Mathematics,  Kail's  Trigonometry. 
Second  i/eaf.—Patoun's  Navigation,  Gregory's  Geometry  and  Fortification  ;  Simson's 

Conic  Sections;  Maclaurin's  and  Emerson's  Fluxions. 
!Z7nrd!i/ear.—Helsham's  Lectures;  Gravesaude;  Cote's  Hydrostatics;  Desaguliera; 

Musclienbroec ;  Keil's  Introduction ;  Martin's  Philosophy,  Maclaurin's  View  of 

Sir  Isaac  Newton's  Philosophy,  Eohault  per  Clarke. 

It  appears  that  the  instruction  was  given  by  lectures,  the  books  of 
which  the  above  is  a  partial  list,  were  (says  Dr.  Smith)  "to  be  con- 
suited  occasionally  in  the  lectures,  for  the  illustrations  of  any  particular 
part ;  and  to  be  read  afterwards,  for  completing  the  whole."  How 
closely  this  advanced  curriculum  of  Dr.  Smith  was  adhered  to,  and  how 
nearly  his  ideal  scheme  came  to  be  realized  in  the  actual  work  of  the 
college,  we  have  no  means  of  determining.  This  much  is  certain,  that 
before  the  Eevolution  the  institution  attracted  a  large  number  of  stu- 
dents. According  to  Dr.  Smith,  the  attendance  in  the  college  alone 
went  as  iiigh  as  one  hundred,  while  the  total  attendance,  including  the 
pupils  of  the  academy  and  charity  schools,  surpassed  three  hundred. 
Of  the  course  of  study  which  he  planned  for  the  institution,  it  has  been 
said  by  competent  judges  that  "no  such  comprehensive  scheme  of  edu- 
cation then  existed  in  the  American  colonies." 

But  there  followed  a  reaction.  Political  troubles  at  the  beginning  of 
the  Eevolutionary  War  broke  up  the  institution.  The  authorities  of 
the  college  were  accused  of  disloyalty,  and  in  1779  the  charter  was  an- 
nulled by  the  Provincial  Assembly,  and  the  college  estate  vested  in  a 
new  board.  Dr.  Smith  was  ejected,  and  in  1791  there  was  organized 
the  "University  of  Pennsylvania."  Many  years  elapsed  before  the 
institution,  regained  the  popularity  it  enjoyed  before  the  war. 

SELF-TAUaHT  MATHEMATICIANS. 

The  mathematicians  mentioned  in  the  previous  pages  were  all  men 
engaged  in  the  profession  of  teaching.  But,  strange  as  it  may  seem, 
the  most  noted  mathematician  and  astronomer  of  early  times  was  not  a 
professor  in  a  college,  nor  had  he  been  trained  within  college  walls. 
We  have  reference  to  David  Eittonhouse.  He  was  born  near  German- 
town.  Pa.,  in  1732.  Until  about  his  eighteenth  year,  he  was  employed 
on  his  father's  farm.  The  advantages  for  obtaining  an  education  in 
rural  districts  were  then  exceedingly  limited,  but  the  elasticity  of  his 
genius  was  superior  to  the  pressure  of  adverse  fortune.  At  the  age  of 
twelve  he  came  in  possession  of  a  chest  of  carpenter's  tools,  belonging 
to  an  uncle  of  his,  who  had  died  some  years  previously.  This  chest 
contained,  besides  the  implements  of  trade,  several  elementary  books 
treating  of  arithmetic  and  geometry.  This  humble  coffer  was  to  him  an 
invaluable  treasure,  for  the  tools  afforded  him  some  means  of  exercising 


38  Teaching  and  history  of  mathematics. 

the  bent  of  his  genius  toward  mechanics,  while  the  books  early  led  his 
mind  to  those  pursuits  for  which  it  was  pre-eminently  fitted.  While  a 
boy  he  is  said  to  have  covered  the  fences  and  plows  on  his  father's  farm 
with  geometrical  figures.  At  the  age  of  seventeen  Le  constructed  a 
wooden  clock. 

The  delicacy  of  his  constitution  and  the  irresistible  bent  of  his  genius 
induced  his  parents  to  yield  to  his  oft-repeated  wish  of  giving  up  farm- 
ing, and  to  procure  for  him  the  tools  of  a  clock  and  mathematical  instru- 
ment maker.  Eittenhouse  now  worked  diligently  with  his  tools  during 
the  day,  and  at  night  spent  a  portion  of  his  time  which  should  have 
been  passed  in  taking  repose  in  the  xDrosecution  of  his  studies.  His 
success  seems  to  have  been  extraordinary,  for  his  biographers  assert 
that  before  the  age  of  twenty  he  was  able  to  read  the  Principia,  and 
that  he  had  discovered  the  method  of  fluxions  without  being  aware  that 
this  had  already  been  done  by  Kewton  and  Leibnitz.  In  Sparks's 
American  Biography  we  read  that  since  Newton  in  his  Principia 
"follows  the  synthetic  method  of  demonstration  and  gives  no  clue  to 
the  analytic  process  by  which  the  truth  of  this  proposition  was  first  dis- 
covered by  him,  *  *  *  Eittenhouse  began  to  search  for  the  instru- 
ment which  might  be  applied  to  the  purpose  of  similar  discoveries,  and 
in  his  researches  attained  the  principles  of  the  method  of  fluxions." 

Dr.  Eush,  in  his  eulogy  on  Eittenhouse,  says  in  the  same  way:  "It 
was  during  the  residence  of  our  ingenious  philosopher  with  his  father 
in  the  country  that  he  made  himself  master  of  Sir  Isaac  Newton's  Prin- 
cipia, which  he  read  in  the  English  translation  of  Mr.  Motte.  It  was 
here,  likewise,  he  became  acquainted  with  the  science  of  fluxions ;  of 
which  sublime  invention  he  believed  himself,  for  a  while,  to  be  the 
author,  nor  did  he  know  for  some  years  afterwards  that  a  contest  had 
been  carried  on  between  Sir  Isaac  Newton  and  Leibnitz  for  the  honor 
of  the  great  and  useful  discovery.  What  a  mind  was  here !  Without 
literary  friends  or  society,  and  with  but  two  or  three  books,  he  became, 
before  he  had  reached  his  four  and  twentieth  year,  the  rival  of  two  of 
the  greatest  mathematicians  in  Europe." 

Our  information  concerning  the  studies  of  our  young  philosopher  is 
so  scanty,  that  we  find  it  impossible  to  determine  the  exact  range  of  his 
thoughts  or  the  consequences  that  flowed  from  them.  Not  the  slight- 
est information  as  to  the  exact  nature  of  his  alleged  invention  has  been 
preserved.  He  himself  seems  to  have  attached  no  weight  to  it.  We 
are  of  the  opinion  that  his  invention,  whatever  it  may  have  been,  was 
not  of  suificient  importance  to  deserve  the  name  of. an  "  invention  of 
fluxions."  If  Eittenhouse  actually  made  an  invention  of  such  trans- 
cending magnitude  before  the  age  of  twenty,  and  at  a  time  when  he 
had  hardly  begun  his  scientific  studies,  how  is  it  that  he  made  not  the 
slightest  approach  to  any  similar  discovery  during  the  forty-four  years 
of  his  maturer  life  ?  Though  always  a  passionate  lover  of  scientific 
pursuits,  he  made  no  original  contributions  whatever  to  the  science  of 


COLONIAL    TIMES.  39 

pure  mathematics.  Science  is  indebteded  to  him  chiefly  for  his  orreries 
and  the  observations  of  the  transit  of  Venus.  We  are,  therefore,  of 
the  opinion  that  the  alleged  invention  of  fluxions  was  little  more  than 
a  "rumor  set  afloat  by  idle  gossip."  It  serves  to  show  us,  however, 
in  what  unbounded  admiration  he  was  held  by  his  countrymen. 

At  the  age  of  nineteen  Eittenhouse  made  the  acquaintance  of  Thomas 
Barton,  a  talented  young  clergyman  who  had  been  a  student  at  the 
University  of  Dublin.  An  intimate  friendship  grew  up  between  them, 
which  proved  "advantageous  to  the  mental  improvement  of  both.  Bar- 
ton was  able  to  furnish  Eittenhouse  with  some  books  suitable  for  his 
instruction.  The  burning  zeal  with  which  our  young  scientist  pursued 
his  studies  appears  from  tlie  following  extract  of  a  letter  he  wrote  to 
Barton  on  September  20,  1756,  at  the  age  of  twenty- four :  "I  have  no 
health  for  a  soldier  [the  country  was  then  engaged  in  war],  and  as  I 
have  no  expectation  of  serving  my  country  in  that  way,  I  am  spending 
my  time  in  the  old  trifling  manner,  and  am  so  taken  with  optics,  that  I 
do  not  know  whether,  if  the  enemy  should  invade  this  part  of  the  coun- 
try, as  Archimedes  was  slain  while  making  geometrical  figures  on  the 
sand,  so  I  should  die  making  a  telescope." 

As  a  mechanic,  Eittenhouse  became  celebrated  for  the  extreme  ex- 
actness and  finish  of  his  workmanship.  Especially  celebrated  were  his 
chronometer  clocks.  It  was  while  thus  engaged  in  the  manufacture  of 
clocks  that  he  planned  and  executed  an  instrument  wbich  brought  into 
play  both  his  mechanical  and  mathematical  skill.  This  instrument  was 
the  orrery.  Concerning  this  wonderful  mechanism,  he  wrote  to  Barton 
January  28,  1767,  as  follows:  "I  do  not  design  a  machine  which  will 
give  the  ignorant  in  astronomy  a  just  view  of  the  solar  system,  but  would 
rather  astonish  the  skilful  and  curious  observer  by  a  most  accurate  cor- 
respondence between  the  situations  and  motions  of  our  little  represent- 
atives of  our  heavenly  bodies  and  the  situations  and  motions  of  those 
bodies  themselves.  I  would  have  my  orrery  really  useful  by  making  it 
capable  of  informing  us  truly  of  the  astronomical  phenomena  for  any 
particular  point  of  time,  which  I  do  not  find  that  any  orrery  yet  made  can 
«do."  It  was,  indeed,  intended  to  be  a  sort  of  a  perpetual  astronom- 
ical almanac,  in  which  the  results,  instead  of  being  exhibited  in  tables, 
were  to  be  actually  exhibited  to  the  eye.  His  orrery  greatly  exceeded 
all  others  in  precision.  It  attracted  very  general  attention  among  well- 
informed  persons,  and  the  Legislature  of  Pennsylvania,  in  appreciation 
of  the  talents  of  Eittenhouse,  voted  that  the  S'lm  of  three  hundred 
pounds  be  given  to  him. 

There-  arose  a  lively  competition  between  different  colleges  in  this 
country  for  the  possession  of  this  orrery.  While  the  College  of  Phila- 
delphia was  negotiating  for  its  purchase,  a  committee  from  the  College 
of  New  Jersey  went  to  examine  it,  and  concluded  to  buy  it  at  once ;  and 
thus,  much  to  the  chagrin  of  Dr.  William  Smith,  Princeton  bore  off  the 
palm  from  Philadelphia  in  obtaining  possession  of  the  first  orrery  con- 


40  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

structed  by  Eittenhouse.  He  afterwards  made  another  one  for  the  Phil- 
adelphia College.  The  author  of  The  Vision  of  Columbus,  a  poem  first 
published  in  1787,  alludes  to  the  Eittenhouse  orrery  m  Philadelphia  and 
the  mass  of  people  crowding  to  the  college  hall  to  see  it,  in  the  following 
lines  (Book  YII) : 

See  the  sage  Eittenhouse,  with  ardeni  eye, 
Lift  the  loEg  tube  and  pierce  the  starry  sky; 
Clear  in  his  view  the  circling  systems  roll, 
And  broader  splendours  gild  the  central  pole. 
He  marks  what  laws  th'  eccentric  wand'rers  bind, 
Copies  Creation  in  his  forming  mind, 
And  bids,  beneath  his  hand,  in  semblance  rise, 
With  mimic  orbs,  the  labours  of  the  skies. 
There  wond'ring  crowds  with  raptur'd  eye  behold 
The  spangled  heav'ns  their  mystic  maze  unfold ; 
While  each  glad  sage  his  splendid  hall  shall  grace, 
With  all  the  spheres  that  cleave  th'  ethereal  space. 

In  August,  1768,  Eittenhouse  was  appointed  by  the  American  Philo- 
sophical Society  in  Philadelphia  as  one  of  a  committee  to  observe  the 
transit  of  Yenus  on  June  3d  of  the  following  year.  A  temporary  ob- 
servatory was  built  by  him  for  the  purpose  near  his  residence  at  Norri- 
ton.  Dr.  William  Smith  aided  him  in  procuring  suitable  instruments, 
and  the  preliminary  arrangements  were  made  with  most  scrupulous 
care.  The  approaching  phenomenon  was  one  of  great  scientific  impor- 
tance. Only  two  transits  of  Venus  had  been  observed  before  his  time, 
and  of  these,  the  first,  in  1639,  had  been  seen  by  only  two  persons. 
These  transits  happen  so  seldom  that  there  cannot  be  more  than  two 
in  one  century,  and  in  some  centuries  none  at  all.  But  the  transits  of 
Venus  are  the  best  means  we  have  for  determining  the  parallax  of  the 
sun.  At  the  approach  of  the  transit,  Eittenhouse  and  his  assistants  in 
this  observation.  Dr.  William  Smith  and  Mr.  Lukens,  then  surveyor- 
general  of  Pennsylvania,  awaited  the  contacts  in  silence  and  anxiety. 
The  observations  were  a  success,  and  established  for  Eittenhouse  tbe 
reputation  of  an  exact  and  careful  astronomer.  The  transit  was  ob- 
served in  Boston  by  Professor  Winthrop,  and  in  Providence  by  Benja- 
min West,  at  almost  ail  the  observatories  in  Europe,  and  in  various 
other  parts  of  the  globe.  During  the  transit  Eittenhouse  saw  one 
phenomenon  which  escaped  the  notice  of  all  other  astronomers.  When 
the  planet  had  advanced  about  half  of  its  diameter  upon  the  sun, 
that  part  of  the  edge  of  the  planet  which  was  off  the  sun's  disc  appeared 
illuminated,  so  that  the  outline  of  the  entire  planet  could  be  seen.  But 
a  complete  circle  of  light  around  Venus  would  indicate  that  more  than 
half  of  Venus  is  illuminated.  This  can  happen,  as  far  as  we  know, 
only  when  the  rays  of  light  are  refracted  by  an  atmosphere.  Hence, 
it  would  follow  from  the  observations  of  Eittenhouse  that  Venus  is  sur- 
rounded, like  the  earth,  by  an  atmosphere.  But  this  appearance  of  a 
ring  of  light  was  not  confirmed  by  other  astronomers,  and  the  state- 


COLONIAL    TIMES.  41 

ment  of  Eittenhouse  excited  no  attention  for  nearly  a  century,  until  liis 
observation  was,  at  last,  conHrmed  by  otBer  astronomers. 

An  important  invention  made  by  Eittenhouse  is  that  of  the  "  colli- 
mator," "  a  device  for  obtaining  a  meridian  mark  without  going  far  away  j 
it  has  lately  come  back  from  Germany,  where  it  was  re-invented."* 

The  reputation  which  Eittenhouse  had  now  acquired  as  an  astronomer 
attracted  the  attention  of  the  Government,  and  he  was  employed  in 
several  important  geodetic  operations.  In  1779  he  was  named  one  of 
the  commissioners  for  adjusting  a  territorial  dispute  between  the  States 
of  Pennsylvania  and  Virginia ;  in  1786  he  was  employed  in  fixing  the 
line  which  separates  Pennsylvania  from  the  State  of  New  York,  and  in 
the  following  year  he  assisted  in  determining  the  boundary  between 
E^ew  York  and  Massachusetts.  In  1791  he  was  chosen  successor  of  Dr. 
Franklin  in  the  presidency  of  the  American  Philosophical  Society.  All 
his  scientific  communications  were  made  to  that  society  and  published 
in  its  Transactions. 

Eittenhouse  came  to  be  looked  up  to  by  Ms  countrymen  as  an  as- 
tronomer equalled  by  few  and  surpassed  by  none  of  his  contemporaries. 
Listen,  if  you  please,  to  Thomas  Jefferson's  estimate  of  him.  In  answer  to 
the  assertion  of  Abbe  Eaynal  that  "America  had  not  yet  produced  one 
good  poet,  one  able  mathematician,  one  man  of  genius  in  a  single  art  or  a 
single  science,"  Jefferson  says :  "When  we  shall  have  existed  as  a  people 
as  long  as  the  Greeks  did  before  they  produced  a  Homer,  the  Eomans  a 
Virgil,  the  French  a  Eacine  and  Voltaire,  the  English  a  Shakespeare 
and  Milton,  should  this  reproach  be  still  true,  we  will  inquire  from  what 
unfriendly  causes  it  has  proceeded,  that  the  other  countries  of  Europe 
and  quarters  of  the  earth  shall  not  have  inscribed  any  name  in  the  roll 
of  poets.  *  *  *  In  war  we  have  produced  a  Washington,  whose 
memory  will  be  adored  while  liberty  shall  have  votaries,  whose  name 
shall  triumph  over  time,  and  will  in  future  ages  assume  its  just  station 
among  the  most  celebrated  worthies  of  the  world.  *  *  *  In  physics 
we  have  produced  a  Franklin,  than  whom  no  one  of  the  present  age  has 
made  more  important  discoveries,  nor  has  enriched  philosophy  with 
fliore,  or  more  ingenious,  solutions  of  the  phenomena  of  nature.  We 
have  supposed  Mr.  BittenJiouse  second  to  no  astronomer  living;  that  in  genius 
he  must  be  the  first,  because  he  is  self-taught.  As  an  artist  he  has  ex- 
hibited as  great  a  proof  of  mechanical  genius  as  the  world  has  ever  pro- 
duced. He  has  not  indeed  made  a  world :  but  he  has  by  iaiitation 
approached  nearer  its  Maker  than  any  man  who  has  lived  from  the 
creation  to  this  day."t 

Such  was  Jefferson's  estimate  of  Eittenhouse.  James  Eenwick  says 
that  "  he  [Eittenhouse]  had  shown  himself  the  equal  in  point  of  learn- 
ing and  skill  as  ah' observer  to  any  j)ractical  astronomer  then  living. " 
Dr.  Bush,  in  his  eulogy,  exclaims :  "What  a  mind  was  here  I  Without 

*The  Development  of  Astronomy  in  the  United  States,  by  Prof.  T.  H.  Safford,  p.  8. 
i  Jeffei-Kon's  Notes  on  Virginia. 


42  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

literary  friends  or  society,  and  with  but  two  or  three  books,  he  became, 
before  he  had  reached  his  foiir-and-twentieth  year,  the  rival  of  two  of 
the  greatest  mathematicians  in  Europe !  " 

If  we  estimate  Kitteuhouse  by  what  he  might  have  done  had  he  had  a 
more  rugged  physical  constitution  and  better  facilities  for  self-develop- 
ment; had  he  had  an  observatory  at  his  disposal  such  as  those  of  his 
great  contemporaries,  Maskelyne  and  William  Herschel  in  England, 
Lalande  and  Count  Oassini  in  France,  Tobias  Mayer  in  Germany,  then 
the  above  estimates  may  be  correct.  But  if  our  astronomer  be  judged 
by  the  original  contributions  which,  under  existing  adverse  circum- 
stances, he  actually  did  make  to  astronomy  and  mathematics,  then  it 
must  be  admitted  that  he  can  not  be  placed  in  the  foremost  rank  of  as- 
tronomers then  living.  Friends  will  judge  him  by  what  he  might  have 
done;  the  world  at  large  will  judge  him  by  what  he  actually  accom- 
plished. Our  greatest  indebtedness  to  Eittenhouse  lies  not  in  the  origi- 
nal contributions  he  made  to  science,  but  rather  in  the  interest  which 
he  aroused  in  astronomical  pursuits,  and  in  the  diffusion  of  scientific 
knowledge  in  the  New  World  which  resulted  from  his  efforts. 

One  who  enjoyed,  in  his  day,  the  reputation  of  being  a  "  great  mathe- 
matician," was  Thomas  Godfrey,  of  Philadelphia.  He  was  a  glazier  by 
trade.  Having  met  accidentally  with  a  mathematical  book,  he  became 
so  delighted  with  the  study  that  by  his  own  unaided  perseverance  he 
mastered  every  book  he  could  get  on  the  subject.  He  pursued  the  study 
of  Latin  in  order  that  he  might  read  Newton's  Principia.  Optics  and 
astronomy  became  his  favorite  studies,  and  the  exercise  of  his  thoughts 
led  him  in  1730  to  conceive  an  improvement  of  the  quadrant.  In  1732 
a  description  of  his  invention  was  sent  to  Dr.  Hadiey  in  England. 
Meantime,  in  1731,  Hadiey  had  made  a  communication  to  the  Koyal 
Society  of  Loudon,  describing  an  improvement  of  the  quadrant  similar 
to  that  of  Godfrey,  The  claims  of  both  parties  were  afterwards  inves- 
tigated by  the  Royal  Society,  and  both  were  entitled  to  the  honor  of  in- 
vention. The  instrument  is  still  called  "  Hadley's  quadrant,"  though  of 
the  two  Godfrey  was  the  first  inventor.  Afterwards  it  appeared  that 
both  had  been  anticipated  in  their  invention  by  Newton. 

Some  of  the  personal  characteristics  of  Godfrey  are  known  to  us 
through  the  writings  of  Benjamin  Franklin.  *'I  continued  to  board 
with  Godfrey,  who  lived  in  part  of  my  house  with  his  wife  and  children, 
and  had  one  side  of  the  shop  for  his  glazier's  business,  though  he  worked 
but  little,  being  always  absorbed  in  mathematics."  In  the  autumn  of 
1727  Franklin  formed  most  of  his  ingenious  acquaintances  into  a  club 
for  mutual  improvement,  called  Junto.  It  met  Friday  evenings.  "  One 
of  the  first  members  of  our  Junto,"  says  Franklin,  "  was  Thomas  God- 
frey, a  self-taught  mathematician,  great  in  his  way,  and  afterwards  in- 
ventor of  what  is  now  called  Hadley's  Quadrant.  But  he  knew  little 
out  of  his  way,  and  was  not  a  pleasing  companion,  as,  like  most  great 
mathematicians  I  have  met  with,  he  expected  universal  precision  in 


Colonial  times.  43 

everything-  said,  and  was  forever  denying  and  distiuguisbing  upon 
trifles,  to  the  disturbance  of  all  conversation." 

This  assertion  of  Franklin  that  all  mathematicians  he  had  met  were 
insufferable  from  their  trifling  and  captious  spirit,  has  been  extensively 
quoted  by  opponents  of  the  mathematical  sciences.  It  was  quoted  by 
Goethe,  and  afterwards  by  Sir  William  Hamilton,  the  metaphysician, 
when  he  was  engaged  in  a  controversy  with  Whewell,  the  celebrated 
author  of  the  History  of  the  Inductive  Sciences,  on  the  educational 
value  of  mathematical  studies.  Hamilton  attempted  to  prove  the  start- 
ling proposition  that  the  study  of  mathematics  not  only  possessed  no 
educational  value,  but  was  actually  injurious  to  the  mind.  He  must 
have  experienced  exquisite  pleasure  in  finding  that  Franklin,  the  great- 
est physical  philosopher  of  America,  had  made  a  statement  to  the  effect 
that  all  mathematicians  he  had  met  were  "  forever  denying  and  dis- 
tinguishing upon  trifles." 

We  shall  not  speak  of  this  controversy,  except  to  protest  against  any 
general  conclusion  being  drawn  from  Franklin's  experience  of  the 
captiousness  of  mathematicians.  Take,  for  examples,  David  Eitten- 
house  and  Nathaniel  Bowditch,  who  were  early  American  mathemati- 
cians, and,  like  Godfrey,  self-taught  men.  Though  Franklin's  state- 
ment may  be  true  in  case  of  Thomas  Godfrey,  it  is  most  positively 
unjust  and  false  when  applied  to  the  other  two  scholars.  The  biogra- 
phers of  David  Rittenhouse  are  unanimous  and  explicit  in  their  assertion 
that  in  private  and  social  life  he  exhibited  all  those  mild  and  amiable 
virtues  by  which  it  is  adorned.  As  to  Nathaniel  Bowditch,  of  whom  we 
shall  speak  at  length  later  on,  we  have  the  reliable  testimony  of  numer- 
ous writers  that  he  was  a  man  remarkable  for  his  social  virtues, 
modest  and  attractive  manners,  and  Franklinian  common  sense. 

Mention  should  be  made  here  of  Benjamin  Banneker,  the  self-taught 
"  negro  astronomer  and  i^hilosopher,"  born  in  Maryland,  who  became 
noted  in  his  neighborhood  as  an  expert  in  the  solution  of  difficult  prob- 
lems, and  who,  with  the  use  of  Mayer's  Tables,  Ferguson's  Astronomy, 
and  Leadbeater's  Lunar  Tables,  made  creditable  progress  in  astronomy, 
and  calculated  several  almanacs.  His  first  almanac  was  for  the  year  1792. 
The  publishers  speak  of  it  as  "having  met  the  approbation  of  several 
of  the  most  distinguished  astronomers  in  America,  particularly  the  cele- 
brated Eittenhouse."  Banneker  sent  a  copy  to  Mr.  Jefferson,  then  Sec- 
retary of  State,  who  said  in  his. reply,  "  I  have  taken  the  liberty  of  send- 
ing your  almanac  to  Monsieur  de  Condorcet,  secretary  of  the  Academy 
of  Sciences  at  Paris,  and  member  of  the  Philanthropic  Society,  because 
I  considered  it  a  document  to  which  your  whole  color  had  a  right  for 
their  justification  against  the  doubts  which  have  been  entertained  of 
them."*  Banneker  was  invited  by  Andrew  Bllicott  to  accompany  "  the 
Commissioners  to  run  the  lines  of  the  District  of  Columbia  "  upon  their 
mission. 

*  History  of  the  Negro  Race  in  America,  by  George  W.  Williams,  p.  386. 


II. 

IKFLUX  OF  EI^TGLISH  MATHEMATICS,  1776-1820. 

The  Eevolutionary  War  bore  down  so  heavily  upon  the  educational 
■work  in  both  elementary  and  higher  institutions,  that  many  of  them, 
for  a  time,  actually  closed  their  doors.  The  majority  of  students  and 
professors  of  Harvard  and  Yale  were  in  the  Army,  or  were  in  some 
other  way  rendering  aid  to  the  national  cause.  The  buildings  of  Nas- 
sau (Princeton)  College  were  for  a  time  used  as  barracks.  The  business 
of  Columbia  College  in  New  York  was  almost  entirely  broken  up.  The 
professors  and  students  of  Eutgers  College  at  New  Brunswick,  N.  J., 
were  sometimes  compelled  by  the  presence  of  the  enemy  to  pursue  their 
academical  studies  at  a  distance  from  New  Brunswick.  The  operations 
of  Brown  University  in  Providence,  R.  I.,  were  discontinued  during 
part  of  the  war,  the  college  bailding  being  occupied  by  the  militia  and 
the  troops  of  Rochambeau.  At  William  and  Mary  College  the  exer- 
cises were  suspended  in  1781  for  about  a  year,  and  the  building  was  oc- 
cupied at  diiferent  times  by  both  British  and  American  troops.  The 
walls  of  the  college  were  "  alternately  shaken  by  the  thunder  of  the  can- 
non at  Yorkfbwn  and  by  the  triumphant  shouts  of  the  noble  bands  who 
had  fought  and  conquered  in  the  name  of  American  Independence." 
Academies  and  primary  schools  were  either  deserted  or  taught  by  wo- 
men and  white-haired  men  too  old  to  fight.  That  the  philosophic  pur- 
isuits  of  scientific  societies  should  have  sunk  very  low  is  not  surprising. 
Fifteen  yeafs  elapsed  between  the  publication  of  the  first  and  second 
volumes  of  the  Transactions  of  the  American  Philosophical  Society  in 
Philadelphia. 

In  spite  of  the  financial  depression  and  poverty  which  existed  imme- 
diately after  the  war,  much  attention  was  paid  to  education.  While  in 
1776  there  existed  in  the  colonies  only  seven  colleges,  the  number  was 
increased  to  nineteen  before  the  close  of  the  eighteenth  century.  Acade- 
mies and  grammar  schools  'were  established,  and  a  large  number  of 
text-books  were  put  through  the  press.  Even  during  the  war  the  print- 
ing-press sent  out  an  occasional  school-book.  Thus,  in  1778,  while  the 
war  was  raging  most  fiercely,  an  edition  of  Dilworth's  spelling-book  was 
printed,  which  contained  in  its  preface  the  following  patriotic  passage ; 
"At  the  beginning  of  the  contest  between  the  Tyrant  and  the  States, 
it  was  boasted  by  our  unnatural  enemy,  that,  if  nothing  more,  they 
could  at  least  shut  up  our  ports  by  their  navy  and  prevent  the  importa- 

44 


INFLUX   OF   ENGLISH  MATHEMATICS.  45 

tion  of  books  and  paper,  so  that  in  a  few  years  we  should  sink  down 
into  barbarity  and  ignorance,  and  be  fit  companions  for  the  Indians, 
our  neighbors  to  the  westward."  These  words,  printed  at  the  darkest 
period  of  the  Revolutionary  War,  disclose  a  spirit  far  from  submissive. 
The  colonists  were  not  quite  ready  to  sink  down  into  barbarity  and 
ignorance.  During  the  twenty-five  years  after  the  Declaration  of  In- 
dependence, more  real  progress  was  made  in  education  than  in  the 
entire  century  preceding.  Between  1776  and  1815  a  large  number  of 
books  on  elementary  and  a  few  on  higher  mathematics  were  published 
in  America.  Many  of  them  were  reprints  of  English  works,  while 
others  were  compilations  by  American  writers,  modelled  after  English 
patterns.  French  and  German  authors  were  almost  unknown.  We 
may  therefore  call  this  the  period  of  the  "  Influs  of  English  Mathe. 
matics"  into  the  United  States.  What  little  mathematics  was  studied 
in  the  colonies  before  the  Eevolution  was,  to  be  sure,  gotten  chiefly 
from  English  sources,  but  the  scientific  currents  thither  were  then  so 
very  feeble  and  slow  that  we  can  hardly  speak  of  an  "  influx." 

Eleiientaet  Schools. 

It  is  a  significant  fact  that  of  the  arithmetics  used  before  the  Eevolu- 
tion, but  one  work  in  the  Euglisli  language  was  written  by  an  American 
author.  It  is  equallj'^  significant  that  with  the  close  of  the  great  strug- 
gle for  liberty,  there  began  a  period  of  activity  in  the  production  of  new 
school-books.  The  second  book  devoted  exclusively  to  arithmetic,  com- 
piled by  an  American  author,  and  printed  in  the  English  language,  was 
the  New  and  Complete  System  of  Arithmetic  by  Mcholas  Pike,  (New- 
buryport,  1788.  )*  "^■^^- — " 

Nicholas  Pike  (1743-1819)  was  a  native  of  New  Hampshire,  graHuated 
at  Harvard  College  in  1766,  and  was  for  many  years  a  teacher  and  after- 
ward a  magistrate  at  Newbury  port  in  Massachusetts.  His  arithmetic 
received  the  approbation  of  the  presidents  and  professors  of  the  leading 
New  England  colleges.  A  recommendation  from  Harvard  professors 
contains  the  following  timely  remark:  "  We  are  happy  to  see  so  use- 
ful an  American  production,  which,  if  it  should  meet  with  the  encour 
agement  it  deserves,  among  the  inhabitants  of  the  United  States,  will 
save  much  money  in  the  country,  which  would  otherwise  be  sent  to 

*  It  appears  that  Greenwood's  Arithmetic,  published  nearly  sixty  years  previously, 
was  at  this  time  not  known  to  exist.  Pike's  Arithmetic  was  called  the ^rs<  American 
work  of  its  kind.  Dr.  Artemas  Martin  has  sent  the  writer  the  American  Antiqua- 
riau,  (Vol.  IV,  No.  12,  New  York,  May,  1888)  giving  an  account  of  Pike's  book.  It 
pives  a  letter  written  by  George  Washington  at  Mount  Vernon,  June  20,  1788,  to 
Nicholas  Pike,  in  which  the  former  politely  acknowledges  the  receipt  of  a  copy  of 
Pike's  Arithmetic.    We  quote  from  the  letter  the  following  passage : 

"  Its  merits  being  established  by  the  approbation  of  competent  judges,  I  flatter 
myself  that  the  idea  of  its  being  an  American  production  and  the  first  of  the  kind  which 
has  appeared,  will  induce  every  patriotic  and  liberal  character  to  give  it  all  the  coun- 
tenance and  patronage  in  his  power. " 


46  TEACHING  ANB   HISTORY   OP  MATHEMATICS. 

Europe,  for  publications  of  this  kind."  Pike's  arithmetic  passed 
throujjh  many  editions,  was  long  the  standard  mathematical  manual  in 
New  Euglaud  schools,  and  formed  the  basis  for  other  arithmetics.  It 
was  a  very  extensive  and  complete  book  for  that  time.  A  large 
proportion  of  the  rules  were  given  without  demonstration,  while  some 
were  proved  algebraically.  In  addition  to  the  subjects  ordinarily  found 
in  arithmetics,  it  contained  logarithms,  trigonometry,  algebra,  and 
conic  sections,  but  these  latter  subjects  were  so  briefly  treated  as  to  pos- 
sess little  value.  After  the  appearance  of  Webber's,  Day's,  and  Farrar's 
Mathematics  for  colleges,  which  elaborated  these  subjects  at  greater 
length,  they  were  finally  omitted  in  the  fourth  edition  of  Pike's  Arith- 
metic, in  1822.     ~. 

In  1788,  when  the  first  edition  appeared,  English  money  was  still  the 
prevalent  medium  of  exchange  in  the  United  States.  To  be  sure,  Fed- 
eral money  was  adopted  by  Congress  as  early  as  1786,  but  previous  to 
1794  there  .was  no  Unite'd  States  coin  of  the  denomination  of  a  dollar.  It 
was  merely  the  money  of  account,  based  upon  the  Spanish  dollar, 
which  had  long  been  in  use  in  this  country.  Cougress  passed  a  law 
organizing  a  mint  in  1793,  but  permitting  the  circulation  of  foreign 
coins  for  three  years,  by  which  time  it  was  believed  the  new  coinage 
would  be  ready  in  sufficient  amount.  When  dollars  and  cents  began 
to  replace  pounds  and  shillings,  it  became  desirable  that  the  Federal 
currency  be  explained  in  arithmetics  and  taught  in  schools.  In  conse- 
quence of  this,  the  sterling  notation  was  changed  to  Federal  in  the 
third  edition  of  Pike's  arithmetic,  which  was  brought  out  in  Boston  in 
1808  by  Nathaniel  Lord.  Similar  changes  were  made  in  other  arith- 
metics.* 

Down  to  the  year  1800,  the  only  arithmetic  written  by  an  American, 
which  enjoyed  widespread  and  prolonged  popular  favor,  was  the  one  of 
Nicholas  Pike.  In  1800  appeared  a  second  successful  arithmetic,  The 
School-master's  Assistant,  by  Nathan  Daboll,  a  teacher  in  New  London 

*  Contemporaneously  with  Pike's  Arithmetic  there  appeared  in  Philadelphia  the 
Elementary  Principles  of  Arithmetic,  by  Thomas  Sarjeant.  This  book,  as  well  as  the 
Federal  Arithmetic,  or  the  Science  of  Numbers  (Philadelphia,  1793),  by  the  same 
author,  had  only  an  ephemeral  reputation.  John  Gough's  Treatise  on  Arithmetic  in 
Theory  and  Practice,  edited  by  Benjamin  Workman  (Boston,  1789),  as  well  as 
Gough's  American  Accountant,  or  School-master's  New  Assistant,  abridged  by  Benja- 
min Workman  and  revised  by  Patterson  (Philadelphia,  1796),  had  a  rather  limited 
circulation.  Nor  did  John  Vinall's  Arithmetic  (Boston,  1792),  enjoy  better  success. 
After  having  been  a  teacher  in  Newburyport  for  seventeen  years,  Vinall  at  last  be- 
came writing-master  in  a  school  in  Boston,  his  native  city.  Ke  is  said  to  have  been 
coarse  in  speech  and,  like  his  book,  unpopular.  Gordon  Johnson  wrote  an  arithme- 
tic (Springfield,  1792),  which  never  had  more  than  a  passing  local  reputation.  Some- 
what more  successful  was  the  Introduction  to  Arithmetic  (Norwich,  Conn.,  1793),  by 
Erastus  Eoot,  a  graduate  of  Dartmouth,  for  several  years  a  teacher,  and  afterward 
an  active  politician  and  member  of  Congress. 

Our  list  of  arithmetics  printed  previously  to  the  year  1800  includes  the  names  of 
several  other  "quaint  and  curious  volumes,"  which,  after  an  ephemeral  reputation, 


INFLUX   OP  ENGLISH   MATHEMATICS.  47 

(born  1750  and  died  1818).  This  work  passed  through  numerous  edi- 
tions. Though  DaboU  had  to  compete  with  Pike's  Abridged  Arithmetic 
and  with  the  celebrated  Scholar's  Arithmetic  of  Daniel  Adams,  it  nev- 
ertheless acquired  an  extensive  popularity.  The  expression,  "  accord- 
ing to  Daboll,"  came  to  be  a  synonym  for  "  mathematical  correctness." 
It  pushed  aside  the  less  favorite  works.  The  main  element  of  popularity 
of  Daboll's  School-master's  Assistant  lay  in  the  fact  that  it  introduced 
Federal  money  immediately  after  the  addition  of  whole  numbers,  and 
showed  how  to  find  the  value  of  goods  therein  immediately  after  simple 
multiplication.  This  arrangement,  says  the  author,  may  be  of  great 
advantage  to  many  who  perhaps  will  not  have  an  opportunity  to  learn 
fractions.  Decimal  fractions  were  wisely  made  to  precede  vulgar  frac- 
tions. In  the  "  Eecollections  "  by  Peter  Parley,  of  the  town  of  Eidge- 
field.  Conn,,  are  found  the  following  interesting  remarks:  "We were 
taught  arithmetic  in  Daboll,  then  a  new  book,  and  which,  being  adapted 
to  our  measures  of  length,  weight,  and  currency,  was  a  prodigious  leap 
over  the  head  of  poor  old  Dilworth,  whose  rules  and  examples  were 
modelled  upon  English  customs.  In  consequence  of  the  general  use  of 
Dilworth  in  our  schools  for  perhaps  a  century,  pounds,  shillings,  and 
pence  were  classical,  and  dollars  and  cents  vulgar  for  several  succeed- 
ing generations.  ^I  would  not  give  a  penny  for  it' was  genteel ;  'I 
would  not  give  a  cent  for  it'  was  plebeian." 

Since  the  adherence  to  pounds  and  shillings  came  to  be  offensive  to 
the  people  of  the  young  republic,  Mr.  Hawley,  in  1803,  undertook  to 
revise  the  work  and  alter  all  the  problems  to  Federal  currency.  He 
'called  the  new  work  "Dilworth's  Federal  Calculator,"  but  after  this 
change  the  book  was  so  completely  different  from  the  original  that  the 
use  of  Dilworth's  name  in  the  title  seemed  hardly  justifiable.  Be  that 
as  it  may,  the  Federal  Calculator  was  not  a  success. 

At  the  beginning  of  the  nineteenth  century  there  were  three  "  great 
arithmeticians  "  in  America,  namely,  Nicholas  Pike,  Kathan  Daboll,  and 

passed  into  forgetfulness,  never  to  be  resurrected  to  memory  except  by  the  curiosity 
of  some  inquiring  lovers  of  "forgotten  lore."  To  the  above  names  we  add:  The 
American  Tutor's  Assistant,  by  John  Todd,  Philadelphia,  third  edition,  1797;  Arith- 
metic by  Zachariah  Jess,  Philadelphia,  1797 ;  American  Arithmetic,  by  David  Coot, 
New  Haven,  Conn.,  1799;  The  Usher,  comprising  arithmetic  in  whole  numbers, 
Federal  money,  mensuration,  surveying,  etc.,  by  Ezekiel  Little,  Exeter,  1799 ;  Usher's 
Arithmetic,  abridged,  by  Ezekiel  Little,  1804  ;  The  American  Accountant,  by  Chaun- 
cey  Lee,  Lansingburg,  1797;  The  American  Accountant,  by  William  Milne,  New 
York,  1797,  in  which,  instead  of  the  answers  to  the  problems,  which  were  usually 
given,  the  author  gave  the  remainders,  after  casting  out  the  nines  from  the  answers. 
A  curious  little  volume  is  the  following :  "  The  Young  Gentleman's  and  Lady's  Assist- 
ant, containing  Geography,  Natural  Philosophy,  Ehetoric,  Miscellaneous,  to  which  is 
added  a  short  and  complete  system  of  Practical  Arithmetic,  wherein  the  money  of 
the  United  States  of  America  is  rendered  easy  to  the  perception  of  youth.  The 
whole  divided  into  small  sections  for  the  convenience  of  aoliools,  by  Donald  Fraser, 
author  of  the  Columbian  Monitor,  New  ¥ork>  1706." 


48  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

Daniel  Adams.  Having  noticed  the  first  two,  we  shall  briefly  speak  of 
the  third.  Daniel  Adams  published  in  1801  the  Scholar's  Arithmetic, 
a  work  which  in  point  of  merit  towers  far  above  the  mass  of  contempo- 
rary school-books.  Adams  was  a  native  of  Massachusetts,  graduated 
at  Dartmouth  College  in  1797,  and  then  became  teacher,  physician,  and 
editor.  He  taught  school  in  Boston  from  1806  to  1813,  then  removed  to 
New  Hampshire,  where  he  afterward  served  as  State  Senator.  Though 
engaged  in  various  lines  of  thought,  arithmetical  studies  were  his  favor- 
ite. He  furnished  the  school-boys'  satchels  not  only  with  the  Scholars' 
Arithmetic,  but  also  with  the  Primary  Arithmetic,  and,  in  1827,  with 
the  New  Arithmetic,  which  passed  through  numerous  editions.  The 
New  Arithmetic  differed  from  the  Scholars'  Arithmetic  in  being  ana- 
lytic instead  of  synthetic  in  treatment.  The  analytic  or  inductive 
method  of  teaching,  introduced  into  Switzerland  by  Pestalozzi,  was 
gaining  ground  rapidly  in  this  country  at  the  beginning  of  the  second 
quarter  of  the  present  century. 

Between  1800  and  1820,  a  large  number  of  arithmetics  sprang  into 
existence.  Most  of  them  enjoyed  only  a  mushroom  popularity.  Among 
the  more  successful  of  the  new  aspirants  to  arithmetical  fame  were  the 
following :  Jacob  Willetts,  of  Poughkeepsie,  N.  Y. ,  the  author  of  the 
Scholar's  Arithmetic,  1812 ;  William  Kinne,  of  Maine,  who  graduated 
at  Tale  in  1804,  and  subsequently  became  teacher  in  his  native 
State  and  the  author  of  A  Short  System  of  Practical  Arithmetic, 
Hallowell,  second  edition,  1807  5  Michael  Walsh,  the  author  of  A  New 
System  of  Mercantile  Arithmetic,  adapted  to  the  commerce  of  the 
United  States,  Newbury  port,  1801;  Stephen  Pike,  whose  arithmetic 
was  published  in  Philadelphia  in  1813 ;  Samuel  Webber's  Arithmetic, 
1810,  which  was  used  chiefly  by  students  preparing  for  Harvard.* 

Our  list  of  arithmetics  published  during  the  first  twenty  years  of  this 
century  is  doubtless  very  imperfect.  Of  the  larger  number  of  publica- 
tions, the  great  majority  had  only  an  ephemeral  reputation.  Excepting 
those  of  Pike,  Adams,  and  Daboll,  hardly  any  have  survived  the  recol- 
lection even  of  the  aged. 

*  Less  widely  used  were  the  following  books:  Jonathan  Grout's  Guide  to  Practical 
Arithmetic,  1802 ;  Caleb  Alexander's  New  System  of  Arithmetic,  Albany,  1802 ;  W. 
M.  Finlay's  Arithmetical  Magazine,  or  Mercantile  Accountant,  New  York,  1803 ; 
James  Noyes's  Federal  Arithmetic,  1804  ;  the  American  School-master's  ARsistant,  by 
Jesse  Guthrie,  of  Kentucky,  Lexington,  1804;  Samuel  Temple's  System  of  Arithmetic 
in  Federal  Currency,  Boston,  1804;  the  Youth's  Arithmetical  Guide,  byTandon  Ad- 
dington  and  Watson,  Philadelphia,  1805 ;  Mathematical  Manual  for  the  Use  of  St. 
Mary's  College  of  Baltimore,  containing  arithmetic  and  algebra,  by  L.  I.  M.  Cherigne, 
Baltimore,  1806 ;  Kimber's  Arithmetic  Made  Easy  for  Children,  second  edition,  1807  ; 
Ballard's  Gauging  Unmasked,  1806 ;  Eobert  Patterson's  Treatise  on  Arithmetic,  Phil- 
adelphia, 1819;  Daniel  Staniford's  Practical  Arithmetic,  Boston,  1818;  George  Fen- 
wick's  Arithmetical  Essay,  Alexandria,  1810;  Compendium  of  Practical  Arithmetic, 
by  Osgood  Carleton,  Boston,  1810  ;  the  American  Arithmetic,  by  Oliver  Welch  of  New 
Hampshire ;  The  Teacher's  and  Pupil's  Assistant,  by  Dale  Tweed  of  northern  New 
York,  1820;  the  Arithmetic  of  Leonard  Loomis ;  The  Columbian  Tutor's  Assistant,  by 
D.  McCurdy,  Washington,  1819 ;  The  First  Lines  of  Arithmetic,  1818,  by  De  Wolf  and 


INFLUX    OF   ENGLISH   MATHEMATICS.  49 

Between  1815  and  1820  a  reform  in  mathematical  teaching  was  in- 
aug'arated  in  this  country.  Foremost  among  the  leaders  in  this  new 
movement  was  John  Earrar  of  Harvard,  who  translated  into  English 
for  the  use  of  colleges  a  number  of  French  works.  The  French  books 
of  that  time  were  far  in  advance  of  the  English.  This  reform  in  the 
teaching  of  the  more  advanced  mathematics  was  accompanied  by  a 
similar  reform  in  arithmetical  teaching.  The  new  ideas  of  Pestalozzi 
were  vigorously  forcing  their  way  from  Switzerland  to  all  parts  of  the 
civilized  world.  Among  the  earliest  fruits  they  bore  in  this  country 
were  the  First  Lessons  in  Arithmetic,  by  Warren  Colburn,  1821.  This 
l^rimer  contained  points  of  great  excellence,  and  it  had  a  sale  such  as 
no  other  arithmetic  ever  had  before. 

We  do  not  now  speak  of  these  reforms,  except  simply  for  the  purpose 
of  marking  the  end  of  an  old  period  and  the  beginning  of  a  new  one. 

Having  enumerated  the  text- books  published  in  the  United  States 
after  the  Eevolution  and  preceding  the  year  1820,  we  shall  now  briefly 
examine  their  contents.  The  leading  characteristics  which  we  observed 
in  Dilworth's  School-master's  Assistant,  are  found  to  exist  in  the  books 
of  this  period.  The  arithmetics  of  this  time  were  little  more  than  Pan- 
dora's boxes  of  ill-formed  rules  to  be  committed  to  memory.  Eeason- 
ing  was  exiled  from  tbe  realm  of  arithmetic,  and  memory  was  made  to 
rule  supreme.  A  science  chiefly  intended  to  cultivate  the  understand- 
ing was  offered  to  the  exercise  merely  of  memory. 

This  banishment  of  demonstration  and  worship  of  memory  did  not, 
I  am  glad  to  say,  originate  in  this  country.  As  already  remarked,  it 
came  from  England.  About  the  middle  of  the  seventeenth  century 
there  arose  in  England  the  commercial  school  of  arithmeticians.  To 
this  school,  says  De  Morgan,  "  we  owe  the  destruction  of  demonstrative 
arithmetic  in  this  country,  or  rather  the  prevention  of  its  growth.  It 
never  was  much  the  habit  of  arithmeticians  to  prove  their  rules ;  and 
the  very  word  proofs  in  that  science,  never  came  to  mean  more  than  a 

Brown,  teachers  in  Hartford,  Conn.;  the  Arithmetic  of  Zachariah  Jes8ofDelaware(?); 
The  Scholar's  Guide  to  Arithmetic,  by  Plinehas  Merrill  of  New  Hampshire  ;  Collec- 
tion of  Arithmetical  Tables,  Hartford,  1812;  Arithmetic  Simplified,  181«,  by  John  J. 
White ;  and  Tbe  Youth's  Guide,  by  Mordecai  Stewart,  Baltimore,  1818 ;  Kev.  John 
White's  Mental  Arithmetic,  Philadelphia,  1818 ;  "  The  American  Youth :  being  a  New 
and  Complete  Course  of  Introductory  Mathematics :  designed  for  the  use  of  Private 
Students,  by  Consider  and  John  Sterry."    "Vol.  1,  Providence. 

Besides  these  American  works  there  were  a  number  of  foreign  books  republished 
in  this  country.  Among  these  are.  The  Tutor's  Guide,  by  Charles  Vyse,  London,  1770, 
which  reached  the  thirteenth  American  edition  in  Philadelphia  in  1806;  A  Complete 
Treatise  on  Arithmetic,  by  Charles  Hutton,  Edinburgh,  1802,  first  American  edition, 
New  York,  1810 ;  A  System  of  Practical  Arithmetic,  by  Rev.  J.  Joice,  London,  1816, 
and  adapted  to  the  commerce  of  the  United  States  by  J.  Walker,  Baltimore,  1819  ; 
The  Scholar's  Guide  to  Arithmetic,  by  John  Bonnycastle,  London,  1786,  Philadelphia, 
1818.  These  English  books  can  hardly  be  said  to  have  excelled  our  American  arith- 
metics ;  nor  did  they  attain  to  any  remarkable  success  in  the  New  World. 
881— liTo.  3— ~4 


50  TEACHING  AND   HISTOEY    OF   MATHEMATICS. 

test  of  the  correctness  of  a  particular  operation,  by  reversing  the  pro- 
cess, casting  out  the  nines,  or  the  like.  As  soon  as  attention  was  fairly- 
diverted  to  arithmetic  for  commercial  purposes  alone,  such  rational  ap- 
plication as  had  been  handed  down  from  the  writers  of  the  sixteenth 
century  began  to  disappear,  and  was  finally  extinct  in  the  work  of 
Cocker  or  Hawkins,  as  I  think  I  have  shown  reason  for  supposing  it 
should  be  called.  From  this  time  began  the  finished  school  of  teachers, 
whose  pupils  ask,  when  a  question  is  given,  what  rule  it  is  in,  and  run 
away,  when  they  grow  up,  from  any  numerical  statement,  with  the  dec- 
laration that  anything  may  be  proved  by  figures — as  it  may,  to  them."* 
Such  is  the  history  of  the  commercial  school  of  arithmeticians  in  Eng- 
land. In  America  this  school  became  firmly  established  wherever 
arithmetic  was  taught.  Thus  the  sins  of  the  early  English  pedagogues 
were  visited  upon  the  children  in  England  and  America  unto  the  third 
and  fourth  generations.  As  late  as  1818,  one  of  our  American  compilers 
of  text-books,  Daniel  Staniford,  actually  stated  in  the  preface,  as  a 
recommendation  to  his  book,  that  "all  mathematical  demonstrations 
are  purposely  omitted,  to  clear  illustrations  of  the  rules  by  easy  exam- 
ples and  such  as  tend  to  j)repare  the  scholar  for  business."  Not  only 
was  this  method  adopted  in  practice,  but  even  advocated  in  theory.  In 
both  English  and  American  arithmetics  the  rules  were  ill-arranged  and 
disconnected.  The  pupil  had  to  learn  a  dozen  rules  which  might  have 
been  reduced  conveniently  to  two  or  three  principles.  The  continuity 
in  the  reasoning  upon  quantities  expressed  by  integers  and  those  ex- 
pressed in  common  or  decimal  fractions  was  often  so  completely  dis- 
guised that  it  became  necessary  to  repeat  the  rules.  Thus  Dilworth 
and  Bonnycastle  give  in  their  arithmetics  three  distinct  rules,  as  fol- 
lows: 

Eule  of  three  for  integers. 

Eule  of  three  for  vulgar  fractions. 

Eule  of  three  for  decimal  fractions. 

Nicholas  Pike,  Daniel  Staniford,  and  John  Yinall  each  give  ^'Eules  of 
interest,"  and  later  on  again  "  Eules  of  interest  by  decimals."  The  re- 
sult of  this  cumbrous  rule-system  is  that  the  scholar  acquires  the  art  of 
solving  problems,  provided  he  knows  what  rule  it  falls  under,  which  is 
not  always  sure  to  be  the  case,  for  the  first  practical  problem  which  will 
arise  may  be  one  requiring  not  one  rule,  but  a  combination  of  rules, 
which  can  therefore  not  be  solved  directly  by  the  rules  in  his  book. 
And  here  he  is  fairly  aground,  for  he  has  no  mastery  of  principles,  but  is 
the  .abject  slave  of  rules.  Such  a  system  of  arithmetic  has  been  very  ap- 
propriately called  cipJiering^  since  intellect  goes  for  nothing  throughout. 

Among  other  features  which  characterize  old  American  arithmetics 
are  the  following: 

(1)  The  total  absence  of  exercises  in  mental  arithmetic. 

*  Arithmetical  Books,  p.  xxi. 


I 


INFLUX   OF   ENGLISH   MATHEMATICS.  51 

(2)  The  meagre  treatment  of  fractions.  The  number  of  exercises  was 
so  very  limited  that  it  was  impossible  for  the  student  to  acquire  a  mas- 
tery of  fractions  without  additional  drill. 

(3)  The  process  of  "cancellation,"  which  shortens  calculations  so 
much,  was  entirely  unknown.  Strange  as  it  may  seem,  it  is  less  than 
fifty  years  since  cancellation  was  introduced  into  our  arithmetics.  One 
of  the  first  books  containing  this  process  was  published  in  1840  by  0. 
Tracy,  in  New  Haven,  entitled  "  A  l!^ew  System  of  Arithmetic,  in  which 
is  explained  and  applied  to  practical  purposes  *  *  *  the  principle 
of  cancelling.    *    *    * " 

(4)  The  system  of  numeration  in  early  American  arithmetics  was  not 
the  French  now  generally  used,  but  the  English,  in  which  the  digits  of 
a  number  are  distributed  in  periods  of  six,  and  consequently  jDroceed  by 
millions.  This  method  was  first  adopted  by  the  Italians.  Lucas  de 
Burgo  gives  it  in  a  work  written  in  1494.  The  method  of  reckoning  by 
three  places,  as  used  in  this  country  and  on  the  Continent,  seems  to 
have  originated  with  the  Spanish. 

(5)  The  subject  which  we  now  call  proportion  was  then  called  the 
"  Eule  of  Three."  It  was  taught  as  a  mere  rule.  The  principle  under- 
lying it  was  ignored  completely.  That  a  proportion  is  the  expression 
of  the  equality  of  two  ratios  was  then  not  even  hinted  at.  This  fact 
goes  to  explain  a  point  which  otherwise  would  seem  mysterious.  If  pro- 
portion is  the  equality  of  ratios,  then  why  was  not  the  usual  symbol 
used  to  express  that  equality  ?  Why  were  four  dots  used  instead  of  the 
two  horizontal  lines?  The  answer  seems  to  be  that  arithmeticians  were 
not  in  the  habit  of  thinking  of  a  proportion  as  the  equality  of  two  ratios. 
A  ratio  was  expressed  by  two  dots,  and  the  four  dots  were  placed  be- 
tween the  ratios  simply  to  disjoint  the  terms,  and  to  show  that  the  sec- 
ond and  third  terms  of  the  proportion  were  not  in  the  same  relation  to 
each  other  as  the  first  and  second  or  third  and  fourth. 

(6)  The  old  arithmetics  contain  two  methods  of  solving  problems 
which  are  but  rarely  given  in  modern  arithmetics.  The  methods  I  re- 
fer to  were  first  used  by  the  Hindoos  on  the  far  off  banks  of  the  Ganges, 
then  borrowed  from  the  Hindoos  by  the  Arabs,  who  transmitted  them 
to  the  Europeans.  They  are  called  the  methods  of  "  single  position  " 
and  "  double  position."  They  teach  us  how  to  resolve  questions  by 
making  one  or  two  suppositions  of  false  numbers,  and  then  making 
corrections  for  the  resulting  errors. 

We  have  seen  that  previous  to  the  year  1820  a  large  number  of  arith- 
metics were  published  in  this  country  j  counting  both  American  and 
foreign,  there  were  to  our  knowledge  over  sixty  different  authors. 
Notwithstanding  this  fact,  the  majority  of  schools  had  an  inadequate 
supply  of  arithmetics.  In  country  schools  especially,  books  were  scarce 
and  of  a  rather  miscellaneous  character,  such  as  had  been  in  families 
perhaps  half  a  century.  Johnnie  Smith  would,  perhaps,  bring  to  school 
a  dilapidated  copy  of  Dil worth's  Arithmetic,  which  had  been  used  once 


52  TEACHING    AND    HISTOEY    OF   MATHEMATICS. 

by  his  father.  His  classmate,  Billy  Brown,  woald  carry  in  his  satchel 
a  copy  of  Nicholas  Pike's  Abridged  Arithmetic  5  the  curly-headed 
Jimmy  Jones  would  express  his  preference  for  Daboll's  School-masiers' 
Assistant,  while  the  majority  of  the  boys  had  no  books  at  all.  When- 
ever the  supply  of  arithmetics  was  insufficient,  manuscript  books  were 
resorted  to.  Arithmetics  were  sometimes  covered  with  sheepskin,  in 
the  economical  expectation  that  they  might  be  made  to  lead  not  only 
one  boy,  but  also  his  younger  brothers  In  succession,  to  the  golden  sci- 
ence of  numbers. 

We  have  nov^  spoken  of  the  most  popular  arithmetics  once  used  in 
this  country.  We  have  also  briefly  examined  their  contents.  Our  next 
task  will  be  to  ascertain,  as  far  as  possible,  the  manner  in  which  arith- 
metic was  taught  in  the  school-room. 

One  of  the  first  inquiries  in  this  connection  is  regarding  the  quality 
of  the  teachers.  The  best  teachers  in  elementary  schools  that  our  fore- 
fathers knew  were  young  students  who  taught  school  for  money  to 
finish  their  courses  in  theology,  medicine,  or  law.  But  this  class  of 
school-teachers  was  not  large  in  early  days.  The  representative  school- 
masters of  by-gone  times  were  the  itinerant  school-masters.  They  were 
mostly  foreigners.  Their  qualifications  seemed  to  be  the  inability  to 
earn  anything  in  any  other  way.  They  were  generally  without  families 
and  had  no  fixed  residence;  they  kept  school  first  in  one  place  and 
then  in  another,  and  wandered  about  homeless.  Many  were  given  to 
drinking  and  gambling.  As  a  class,  their  knowledge  was  limited  to 
the  merest  elements.  We  are  told  that  as  late  as  1822,  in  a  town  in 
the  State  of  Connecticut,  six  out  of  fifteen  applicants  for  positions  as 
teachers  were  rejected  because  they  did  not  understand  notation  and 
numeration  of  numbers.  And  yet  these  candidates  came  well  recom- 
mended as  having  taught  school  acceptably  in  other  towns  for  one,  two, 
or  three  winters.  If  this  story  be  true,  then  it  will  not  seem  strange  to 
hear  that  it  was  a  common  practice  for  teachers  in  those  early  days  to 
have  their  scholars  "  skip"  fractions.  This  omission  was  justified  on 
the  ground  that "  fractions  were  rarely  used  in  business,"  but  there  were 
generally  other  good  and  untold  reasons  for  "  skipping  "  the  subject. 
There  were  few  schools  that  carried  the  students  beyond  the  rule  of 
three  or  proportion. 

We  have  seen  the  great  defects  in  the  old  arithmetics.  The  state- 
;'  ment  of  rules  took  the  place  of  explanations  and  reasoning.  If  the 
school-masters  had  been  competent  and  well  trained,  then  the  defects 
of  bad  books  might  have  been  remedied  by  skillful  teaching,  but  the 
teaching  was  generally  of  the  poorest  kind.  The  truth  of  this  asser- 
tion will  be  strikingly  illustrated  by  a  few  examples.  Joseph  T.  Buck- 
ingham tells  us  how,  in  1790  or  1791,  when  he  was  about  twelve  years 
old,  he  began  to  learn  arithmetic.  I  quote  his  exact  words  :  "  I  told 
the  master  I  wanted  to  learn  to  cipher.    He  set  me  a  sum  in  simple  ad- 


INFLUX    OP   ENGLISH   MATHEMATICS.  DO 

dition,  five  columns  of  figures  and  six  figures  in  each  column.  All  the 
instruction  he  gave  me  was— add  the  figures  in  the  first  column,  carry 
one  for  every  ten,  and  set  the  overplus  down  under  the  column.  I  sup- 
posed he  meant  by  the  first  column  the  left-hand  column,  but  what  he 
meant  by  carrying  one  for  every  ten  was  as  much  a  mystery  as  Sam- 
son's riddle  was  to  the  Philistines.  I  worried  my  brains  an  hour  or 
two,  and  showed  the  master  the  figures  I  had  made.  You  may  judge 
what  the  amount  was  when  the  columns  were  added  from  left  to  right. 
The  master  frowned  and  repeated  his  former  instruction — add  up  the 
column  on  the  right,  carry  one  for  every  ten,  and  set  down  the  remainder. 
Two  or  three  afternoons  (I  did  not  go  to  school  in  the  morning)  were 
spent  in  this  way  when  I  begged  to  be  excused  from  learning  to  cipher, 
and  the  old  gentleman  with  whom  I  lived  thought  it  was  time  wasted ; 
and  if  I  attended  the  school  any  further  at  that  time  reading,  spelling, 
and  a  little  writing  were  all  that  was  taught."  Next  winter  a  more 
communicative  teacher  had  the  school,  "  and  under  him  some  progress 
was  made  in  arithmetic,  and  I  made  tolerable  acquisition  in  the  first 
four  rules,  according  to  Dilworth's  School-master's  Assistant,  of  which 
the  teacher  and  one  of  the  oldest  boys  had  a  copy." 

An  experience  similar  to  that  of  the  writer  just  quoted  was  that  of 
Warren  Burton.  In  his  book  entitled  "  The  District  School  as  it  was, 
by  one  who  went  to  it,"  he  says  that  simple  addition  was  easy ;  "  but 
there  was  one  thing  I  could  not  understand — that  carrying  of  tens.  It 
was  absolutely  necessary,  I  perceived,  in  order  to  get  the  right  answer, 
yet  it  was  a  mystery  which  that  arithmetical  oracle,  our  school-master, 
did  not  see  fit  to  explain.  It  is  possible  that  it  was  a  mystery  to  him. 
Then  came  subtraction  5  the  borrowing  of  ten  was  another  unaccount- 
able operation.  The  reason  seemed  to  be  then  at  the  very  bottom  of 
the  well  of  science  5  and  there  it  remained  for  that  winter,  for  no  friendly 
bucket  brought  it  up  to  my  reach." 

Mr.  William  B.  Fowle  gives  an  interesting  account  of  John  Tileston, 
who  was  chief  writing-master  in  a  reading  school  in  Boston  about  the 
year  1790.  It  illustrates  both  the  modes  of  teaching  and  the  compe- 
tency of  teachers.  One  regulation  of  that  school  required  the  writing- 
master  to  teach  "  writing,  arithmetic,  and  the  branches  usually  taught 
in  town  schools,  including  vulgar  and  decimal  fractions."  Mr.  Fowle 
speaks  of  Tileston  as  follows :  * 

"  He  loved  routine.  *  *  *  Printed  arithmetics  were  not  used  in 
the  Boston  schools  until  after  the  writer  left  them,  and  the  custom  was 
for  the  master  to  write  a  problem  or  two  in  the  manuscriiDt  of  the  pupil 
every  other  day.  No  boy  was  allowed  to  cipher  till  he  was  eleven  years 
old,  and  writing  and  ciphering  were  never  performed  on  the'same  day. 
Master  Tileston  had  thus  been  taught  by  Master  Proctor  [his  predeces- 
sor], and  all  the  sums  he  set  for  his  pupils  were  copied  exactly  from  his 

•  Barnard's  Journal,  Vol.  V,  p.  336. 


54  TEACHING   AND  HISTORY   OP   MATHEMATICS. 

old  manuscript.  Any  l)oy  could  copy  the  work  from'the  manuscript  of 
any  other  further  advanced  than  himself,  and  the  writer  never  heard  of 
any  explanation  of  any  principle  of  arithmetic  while  he  was  at  school. 
Indeed,  the  pupils  believed  that  the  master  could  not  do  the  sums  he 
set  for  them.  *  *  *  It  is  said  that  a  boy  who  had  done  the  sum 
set  for  him  by  Master  Tileston  carried  it  up,  as  usual,  for  examination. 
The  old  gentleman,  as  usual,  took  out  his  manuscript,  compared  the 
slate  with  it,  and  pronounced  it  wrong.  The  boy  went  to  his  seat  and 
reviewed  his  work,  but  finding  no  error  in  it,  returned  to  the  desk,  and 
asked  Mr.  Tileston  to  be  good  enough  to  examine  the  work,  for  he  could 
find  no  error  in  it.  This  was  too  much  to  require  of  him.  He  growled, 
as  his  habit  was  when  displeased,  but  he  compared  the  sums  again,  and 
at  last,  with  a  triumphant  smile,  exclaimed, '  See  here,  joi\  ma-ly  (gnarly) 
wretch,  you.  have  got  it,  ''  If  four  tons  of  hay  cost  so  much,  what  will 
seven  tons  cost  ?  "  when  it  should  be,  "  If  four  tons  of  Englisli  hay 
cost  so  and  so."  Now  go  and  do  it  all  over  again.'"  In  this  story,  it 
may  be  remarked,  some  allowance  must  doubtless  be  made  for  the 
genius  of  the  narrator. 

The  illustrations  which  have  been  given  of  the  incompetency  of 
teachers  may  appear  to  be  exaggerations,  and  we  certainly  wish  that 
for  the  good  name  of  our  early  educators  they  were  exaggerations. 
But  the  more  we  inquire  into  this  subject  and  the  more  evidence  we 
accumulate,  the  stronger  the  conviction  becomes  that  most  of  them  are 
not  exaggerations,  but  fair  samples  of  the  teaching  done  by  the  aver- 
age school-master  in  elementary  schools  eighty  or  one  hundred  ypars 
ago. 

In  view  of  these  facts,  the  most  obstinate  pessimist  will  be  forced  to 
admit  that,  within  the  last  one  hundred  years,  progress  has  been  made. 
We  have  better  books  and  abler  teachers.  Our  methods  of  teaching 
arithmetic,  though  still  imperfect,  are  a  prodigious  leap  in  advance  of 
those  of  olden  times.  We  boast  of  our  material  progress,  and  we 
certainly  have  great  reason  for  doing  so,  but  the  progress  in  intellect- 
ual fields,  and  in  education  in  particular,  though  less  ostentatious,  is ' 
none  the  less  instructive.  _ 

Not  without  interest  are  the  following  two  stanzas  of  a  poem,  en- 
titled "A  Country  School,"  which  was  anonymously  contributed  to  the 
New  Hampshire  Spy,  and  preserved  in  E.  H.  Smith's  collection  of 
American  poems  (1793) : 

Will  pray  Sir  Master  mend  my  pen  ? 

Say,  Master,  that's  enough.     Here,  Ben, 

Is  this  your  copy  ?    Can't  you  tell  ? 

Set  all  j^our  letters  parallel. 

I've  done  my  sum — 'tis  just  a  groat — 

Let's  see  it.     Master,  m'  I  g'out  ? 

Yes,  bring  some  wood  in.     What's  that  noise? 

Jt  isnH  I,  sir,  it's  them  loya. 


INFLUX    OF   ENGLISH   MATHEMATICS.  6o 

Come  Billy,  read.    What's  tliat  ?  That's  A— 
Sir,  Jim  has  snatch'd  my  rule  away — 
Eeturn  it,  James.    Here  rule  witli  this — 
Billy,  read  on — that's  crooked  S. 
Read  iu  the  spelling-book.     Begin — 
The  boys  are  out.    Then  call  them  in — 
My  nose  Meeds,  mayn't  I  get  some  ice, 
And  hold  it  in  my  breeches?    Yes. 
John,  keep  your  seat.    My  sum  is  more- 
Then  do't  again — divide  by  four,  * 
By  twelve,  and  twenty — mind  the  rule. 
Now  speak,  Manasseh,  and  spell  tool. 
I«au't.     Well  try.     T,  W,  L. 
Not  wash'd  your  hands  yet,  booby,  ha? 
You  had  your  orders  yesterday. 
.   Give  me  the  ferule,  hold  your  hand, 
•     Oh !  Oh !  there — mind  my  next  command. 

Colleges. 

Before  proceeding  to  the  history  of  mathematics  in  higher  institutions, 
we  shall  speak  of  American  reprints  of  English  mathematical  works  for 
colleges.  First  of  all  comes  that  good  old  Greek  geometry  of  Euclid, 
of  which  the  English  made  excellent  translations.  An  edition  of  Euclid 
appeared  in  Worcester  in  1784.  This  seems  to  be  the  earliest  American 
edition.  After  the  beginning  of  this  century  numerous  editions  of  it 
were  published.  In  1803,  Thomas  and  George  Palmer,  in  Philadelphia, 
published  Robert  Simson's  Euclid,  together  with  the  book  of  Euclid's 
Data,  and  the  Elements  of  Plane  and  Spherical  Trigonometry.  The  book 
was  sold  "  at  the  book-stores  in  Philadelphia,  Baltimore,  Washington, 
Petersburg,  and  Norfolk." 

Prof.  S.  Webber  says,  in  his  "  Mathematics,"  that  a  good  American 
edition  of  Play  fair's  Elements  of  Euclid,  containing  the  first  six  books, 
with  two  books  on  the  geometry  of  solids,  was  given  by  Mr.  F.  Nicholls, 
of  Philadelphia,  1806.  John  D.  Craig,  teacher  of  mathematics  in  Balti- 
more, brought  out  an  edition  of  Euclid  in  1818.  Eobert  Simson's  Eu- 
clid was  republished  in  Philadelphia  in  1821 ;  Playfair's  in  New  York 
in  1819  and  1824,  and  in  Philadelphia  in  1826.  In  1822  appeared  the 
following  work  :  "  Euclid's  Elements  of  Geometry,  the  first  six  books,  to 
which  are  added  the  Elements  of  Plane  and  Spherical  Trigonometry,  a 
System  of  Conic  Sections,  Elements  of  Natural  Philosophy  as  far  as  it 
relates  to  Astronomy,  according  to  the  Newtonian  System,  and  Elements 
of  Astronomy,  with  notes  by  Eev.  John  Allen,  A.  M.,  professor  of  mathe- 
matics at  the  University  of  Maryland."  John  D.  Craig,  in  a  notice  of 
this  book,  says  that  Newton's  work  at  this  day  is  "almost  a  locked 
treasure  among  us,"  owing  to  the  "  scarcity  of  tracts  giving  the  necessary 
preparatory  knowledge."  The  object  of  this  volume  was  to  supply  that 
want. 

An  English  mathematician,  whose  works  found  their  way  across  the 
ocean,  was  John  Bonnycastle,  professor  of  mathematics  at  the  Eoyal 


56  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

Military  Academy,  Woolwich.  His  Introduction  to  Algebra  (London, 
1782)  was  revised  and  edited  in  this  country  by  James  Eyan  in  1822. 
Bonnycastle  was  a  teacher  of  rules  rather  than  principles. 

An  English  author  well-known  in  this  country  was  Thomas  Simpson. 
His  Treatise  on  Algebra  was  published  in  Philadelphia  in  1809.  The 
second  American  from  the  eighth  English  edition,  revised  by  David 
McClure,  teacher  of  mathematics,  came  out  in  Philadelphia  in  1821. 
As  was  frequently  the  case  in  those  days,  all  demonstrations  are  here 
given  by  themselves  in  the  manner  of  notes  placed  below  a  horizontal 
line  on  the  page.  They  could  be  taken  or  omitted  by  teacher  and  pupil 
at  pleasure,  and  were  generally  omitted.  The  author's  explanations 
and  demonstrations  wanted  simplicity,  and  we  need  not  wonder  if  they 
were  "  looked  upon,  by  some,  as  rather  tending  to  throw  new  difficulties 
in  the  way  of  the  learner  than  to  the  facilitating  of  his  progress." 

Another  English  algebra  reprinted  here  was  that  of  B.  Bridge,  fellow 
of  St.  Peter's  College,  Cambridge  (second  American  edition  from 
eleventh  London,  Philadelphia,  1839).  We  are  informed  that  this  work 
was  introduced  into  the  University  of  Pennsylvania,  the  Western  Uni- 
versity, Pittsburg,  in  Gummere's  School  at  Burlington,  the  Friends' 
College  at  Haverford,  and  "  a  great  number  of  the  best  schools  in  the 
United  States."  The  Three  Conic  Sections,  by  the  same  author,  was 
also  patronized  by  some  of  our  colleges.  This  subject  was  here  treated 
purely  synthetically,  as  was  the  case  also  in  Eobert  Simson's  Conic 
Sections,  reprinted  here  in  1809  (?),  and  in  all  other  English  treatises  of 
that  time.  Analytic  methods,  which  proved  so  powerful  in  the  hands 
of  mathematicians  on  the  Continent,  were  still  underrated  in  England. 
The  exclusive  adherence  to  the  synthetic  method  was  due  to  an  exces- 
sive worship  of  the  views  of  Kewton,  who  favored  synthesis  and  em- 
ployed it  throughout  his  Principia. 

We  next  mention  Eev.  Samuel  Vinoe's  Fluxions,  printed  in  Philadel- 
phia in  1812,  or  about  twenty  years  after  its  first  appearance  in  Eng- 
land. This  seems  to  be  the  only  work  devoted  exclusively  to  fluxions 
which  was  ever  published  in  this  country.  Before  the  introduction  of 
the  Leibnitzian  notation  it  was  the  text-book  most  generally  used  in 
our  colleges,  whenever  fluxions  were  taught.  An  edition  of  Vince's 
Astronomy  came  out  in  Philadelphia  in  1817.  Of  his  other  works,  his 
''Conic  Sections,  as  preparatory  to  the  reading  of  Newton's  Principia," 
was  best  known  in  America.  Yince  held  the  position  of  Plumian  pro- 
fessor of  astronomy  and  experimental  philosophy  in  the  University  of 
Cambridge,  England.  His  works  generally  lacked  elegance,  and  failed 
to  teach  the  more  modern  and  improved  forms  of  the  mathematical 
science. 

More  prominent  than  any  of  the  English  authors  here  mentioned  was 
Charles  Hutton.  His  Course  of  Mathematics  was  edited  in  America 
by  Eobert  Adrain,  and  will  be  spoken  of  again  later. 


INFLUX    OF   ENGLISH   MATHEMATICS.  67 

HARVARD   COLLEGE. 

It  has  already  been  stated  that  the  chair  of  mathematics  and  natural 
philosophy  at  Harvard  was  occupied  from  1779  to  1788  by  Eev.  Samuel 
Williams,  a  pupil  of  John  Winthrop.  Williams  wrote  manuscript  books 
on  astronomy,  mathematics,  and  philosoi)hy.  His  mathema-tical  manu- 
scripts were  probably  studied  in  place  of  Ward's  Mathematics,  which 
had  been  used  by  his  predecessor,  John  Winthrop.  We  possess  hardly 
any  information  on  the  mathematical  instruction  during  his  time.  The 
following  is  taken  from  the  diary  of  a  student  who,  in  178G,  applied  for 
admission  to  the  third  term  of  the  Junior  year:  "  Mr.  Williams  asked 
me  if  I  had  studied  Euclid  and  arithmetic."  *  This  question  having 
been  answered,  apparently,  in  the  af&rmative,  he  was  admitted.  From 
this  it  would  seem  that  at  that  time  Euclid  and  arithmetic  were  the 
only  mathematical  studies  pursued  previous  to  the  close  of  the  Junior 
year.  The  fourth  year,  says  Amory,  seems  to  have  been  principally 
occupied  in  the  study  of  mathematics.  From  a  quotation  given  by 
Amory,  we  infer  that  algebra  was  a  college  study  at  this  time.  It  had 
probably  been  so  during" the  last  fifty  years,  but  we  possess  no  data 
from  which  this  could  be  positively  affirmed. 

A  ray  of  light  upon  the  inner  workings  of  the  college  is  thrown  by 
quotations  from  the  diary  of  a  student  who  was  at  Harvard  in  1786. 
They  show  that  the  tutors  of  the  college  failed  to  command  the  esteem 
-tind  respect  of  the  students.  Complaints  were  made  that  the  Greek 
tutor  was  too  young.  "  Before  he  took  his  second  degree,  which  was 
last  commencement,  he  was  chosen  a  tutor  of  mathematics,  in  which  he 
betrayed  his  ignorance  often."  Of  another  tutor  it  is  remarked :  "  We  re- 
cite this  week  to  our  own  tutor  in  Gravesand's  Experimental  Philosophy. 
This  gentleman  is  not  much  more  popular  than  the  rest  of  the  tutors." 
Whoever  has  observed  the  freedom  with  which  college  boys  speak  of 
their  instructors,  knows  that  statements  like  these  must  be  taken  with 
some  allowance.  But  the  practice  alluded  to  above,  of  selecting  grad- 
uates who  had  excelled  mainly  in  classical  studies  as  tutors  in  math- 
ematics, seems  absurd.  And  yet  it  is  well  known  that  this  custom  was 
continue,d  even  in  some  of  our  best  colleges  down  to  a  comparatively 
recent  date.  The  objections  to  the  custom  which  existed  at  Harvard 
previous  to  1767  are  still  more  obvious.  In  the  early  days  of  Harvard 
each  tutor  taught  all  branches  to  the  class  assigned  to  him,  throughout 
the  whole  collegiate  course.  But  in  1767  the  rule  was  introduced  that 
one  tutor  should  not  teach  all  the  subjects,  but  only  one  subject,  such 
as  Greek;  another  tutor  should  have  Latin;  another  mathematics, 
physics,  natural  philosophy,  etc. 

In  1789  Samuel  Williams  was  succeeded  in  the  professorship  of 
mathematics  and  natural  philosophy  by  Samuel  Webber.  Webber  en- 
gaged, while  a  boy,  in  agricultural  pursuits,  and  at  the  advanced  age  of 

*  "  Old  Ctambridge  and  New,"  by  Amory,  in  North  American  Eeview,  Vol.  114,  1872. 


58  TEACHING   AND    HISTORY   OF   MATHEMATICS. 

twenty,  in  1780,  he  entered  Harvard  College.  After  graduating  he  re- 
mained two  years  at  the  college,  studying  theology.  He  then  held  a 
tutorship  till  his  appointment  to  the  HoUis  professorship,  in  which 
office  he  spent  seventeen  of  the  most  important  years  of  his  life.  In 
1800  he  was  elected  president  of  Harvard.  He  occupied  this  position 
till  his  death,  in  1810.  Henry  Ware,  in  his  eulogy  of  Webber,  says : 
"  As  a  scholar  his  attainments  were  substantial,  embracing  various 
branches  of  learning,  but,  mathematical  science  being  most  congenial 
to  his  taste  and  habits,  he  quitted  his  professorship  for  the  presidency 
with  reluctance.  In  communicating  instruction,  he  united  patience  and 
facility  with  a  thorough  acquaintance  with  his  subject."  Edward  Ever- 
ett, who  was  a  student  at  Harvard  in  Webber's  time,  makes  a  some- 
what different  estimate  of  him,  saying  that  Webber  was  "reputed  a 
sound  mathematician  of  the  old  school,  but  rather  too  much  given  to 
routine."*  In  another  place,  Everett  speaks  of  him  in  the  same  way  as 
"  a  person  of  tradition  and  routine."  Judge  Story  says  of  him,  "  Pro- 
fessor Webber  was  modest,  mild,  and  quiet,  but  unconquerably  reserved 
and  staid."  t 

In  1787,  just  before  Samuel  Webber  was  elected  ijrofessor,  the  course 
of  studies  at  Harvard  was  revised,  with  a  view  of  raising  the  standard 
of  learning.  According  to  the  new  scheme,  the  classics  "formed  the 
principal  study  during  the  first  three  college  years.  The  Freshmen 
were  instructed,  also,  in  rhetoric,  the  art  of  speaking,  and  arithmetic, 
the  Sophomores  in  algebra,  and  other  branches  of  mathematics ;  the 
Juniors  in  Livy,  Doddridge's  Lectures,  and  once  a  week,  the  Greek  Tes- 
tament ;  the  Seniors  in  logic,  metaphysics,  and  ethics."| 

The  elementary  mathematics  were  now  studied  in  the  first  half  of  the 
college  course  instead  of  the  second  half.  The  Freshmen  and  Sopho- 
mores now  began  taking  mathematics,  though,  we  fear,  only  in  ineffect- 
ual homoeopathic  doses.  If  arithmetic  was  begun  in  the  Freshman  year, 
then  we  may  be  sure  that  no  very  extensive  course  could  have  been 
given  before  the  close  of  the  Sophomore  year.  According  to  Judge 
Story,  Saunderson's  Algebra  was  used  in  1795  or  1796.  The  original 
work  of  this  blind  mathematician  was  extensive,  and  in  two  volumes. 
The  book  used  at  Harvard  consisted  most  likely  of  "  Selected  parts  of 
Professor  Saunderson's  Elements  of  Algebra,"  published  in  one  vol- 
ume.§ 

W.  Williams,  a  classmate  of  Channing  (class  of  1798),  says:  "The 
Sophomore  year  gave  us  Euclid  to  measure  our  strength.  Many  halted 
at  the  ^]^ons  asinorum.^  But  Channing  could  go  over  clear  at  the  first 
trial,  as  could  some  twelve  or  fifteen  of  us.  This  fact  is  stated  to  show 
that  he  had  a  mind  able  to  comprehend  the  abstrusities  of  mathematics, 

*  Old  Cambridge  and  New,  Vol.  IV,  p.  199. 

t  Memoir  of  W.  E.  Channing,  by  W.  F.  Channing,  fourth  ed.,  1850,  Vol.  I,  p.  47. 

t  Quincy's  History  of  Harvard  University,  Vol.  II,  p.  279, 

{  Third  edition,  London,  1771. 


INFLUX   OF   ENGLISH   MATHEMATICS.  59 

though,  to  my  apprehension,  he  excelled  more  decidedly  in  the  Latin 
and  Greek  classics,  and  had  a  stronger  inclination  to  polite  literature." 

We  are,  moreover,  told  of  Ohanning  :  "  He  delighted,  too,  in  geom- 
etry, and  felt  so  rare  a  pleasure  in  the  perception  of  its  demonstrations 
that  he  took  the  fifth  book  of  Euclid  with  him  as  an  entertainment  dur- 
ing one  vacation."  Such  experiences  are  freguent  with  a  student  of 
advanced  mathematics,  but,  unfortunately,  too  rare  with  pupils  study- 
ing the  elements. 

If  the  course  given  by  Quincy  be  supposed  complete,  then  no  mathe- 
matics was  studied  after  the  Sophomore  year.  This  was  probably  not 
>rue ;  it  certainly  was  not  true  ten  years  later.  In  1797,  at  least  soma 
of  the  students  pursued  the  more  advanced  mathematics  during  the 
latter  part  of  the  college  course.  A  glimpse  of  light  on  this  subject  is 
thrown  by  the  following  quotation  from  the  eulogy  of  John  Pickering: 
'^  Great  as  was  his  enthusiasm  for  classical  learning,  he  had  in  college 
as  real  a  love  for  the  study  of  mathematics,  and  highly  distinguished 
himself  in  this  department.  N'ear  the  close  of  his  Senior  year  he  re- 
ceived the  honor  of  a  mathematical  part,  which  appeared  to  give  him 
more  pleasure  than  all  his  other  college  honors.  It  afforded  him  an 
opportunity  to  manifest  his  profound  scholarship  in  a  manner  most 
agreeable  to  his  feelings.  When  he  had  delivered  the  corporation  and 
overseers  this  part,  containing  solutions  of  problems  by  fluxions,  he 
h^d  the  rare  satisfaction  to  be  told  that  one  of  them  was  more  elegant 
than  the  solution  of  the  great  Simpson,  who  wrote  a  treatise  on  flux- 
ions, in  which  the  same  problem  was  solved  by  him."  It  follows  from 
this  that  provisions  were  made  for  the  study  of  fluxions,  at  least  for 
students  who  may  have  desired  to  study  them. 

Of  the  mathematical  theses,  written  by  Juniors  and  Seniors,  which 
have  been  deposited  in  the  Harvard  Library,  one  hundred  and  thirty- 
three  were  written  during  the  period  from  1781  to  1 807.  Of  these,  the 
great  majority  are  on  the  calculation  and  projection  of  eclipses.  Sur- 
veying and  the  algebraic  solution  of  problems  receive  also  a  large 
amount  of  attention.  Of  the  one  hundred  and  thirty-three  theses  only 
seven  show  by  their  titles  that  they  contain  "fluxionary  problems."  Their 
dates  are,  1796,  1803  (two),  1804,  and  1806  (three).  After  1807,  theses 
containing  solutions  of  problems  on  fluxions  are  quite  numerous.  It 
may  be  of  interest  to  state  that  John  Farrar,  the  future  professor  of 
mathematics  at  Harvard,  wrote  in  1803  a  thesis  on  the  "Calculation 
and  Projection  of  a  Solar  Eclipse."  James  Savage,  the  great  authority 
on  American  genealogy,  furnished  a  colored  view  of  churches  and  col- 
lege buildings;  Everett,  the  diplomatist,  a  colored  "  Templi  Upiscopalis 
Belineatio  PerspecUimP  One  thesis  is  on  the  "Calculation  and  Pro- 
jection of  a  Solar  Eclipse  which  took  place  in  the  year  of  the  Cruci- 
fixion."* 

*  Biographical  Contributions  of  tlie  Library  of  Harvard  University,  No.  32. 


60  TEACHING   AND    HISTOEY    OF   MATHEMATICS. 

In  1802  the  standard  for  admission  to  Harvard  College  was  raised. 
In  matliematics  a  knowledge  of  Arithmetic  to  tlie  "Eule  of  Three"  was 
required.  Thus,  in  1803,  lor  the  first  time  had  it  become  necessary,  ac- 
cording to  regulations,  for  a  boy  to  know  something  about  arithmetic 
before  he  could  enter  Harvard.  We  surmise,  however,  that  the  require- 
ments in  arithmetic  were  very  light,  for  we  know  from  the  diary  of  a 
student  in  the  Freshman  class  in  1807  that  arithmetic  continued  to  be 
taught  during  the  first  year  at  college.*  After  1816  the  whole  of  arith- 
metic was  required  for  admission. 

From  the  beginning  of  the  nineteenth  century  on,  we  can  get  more 
definite  informatiom  regarding  the  extent  to  which  the  mathematical 
studies  were  pursued.  We  need  only  examine  the  college  text-books 
which  began  then  to  be  printed  in  this  country.  The  earliest  mathematical 
text-books  for  colleges,  written  by  an  American  author,  are  those  of  Sam- 
uel Webber.  In  1801  were  published  in  two  volumes  his  "  Mathematics, 
compiled  from  the  best  authors  and  intended  to  be  a  text-book  of  the 
course  of  private  lectures  on  these  sciences  in  the  University  of  Cam- 
bridge." A  second  edition  appeared  1808.  These  works  were  for  a 
time  almost  exclusively  used  in  New  England  colleges,  but  they  finally 
gave  place  to  translations  from  French  works,  executed  by  John  Far- 
rar,  the  successor  of  Webber  in  the  professorship  of  mathematics. 

Within  two  volumes,  each  of  460  pages  and  in  large  print,  are  em- 
braced the  following  subjects :  arithmetic,  logarithms,  algebra,  geome- 
try, plane  trigonometry,  mensuration  of  surfaces,  mensuration  of  solids, 
gauging,  heights  and  distances,  surveying,  navigation,  conic  sections, 
dialing,  spherical  geometry,  and  spherical  trigonometry.  Some  idea  of 
the  extent  to  which  each  branch  of  mathematics  was  carried  may  be  ob- 
tained, if  we  state  that  in  Webber's  works  124  pages  were  given  to 
algebra,  while  Newcomb's  Algebra,  for  instance,  numbers  545  pages. 
The  subject  of  conic  sections  was  disposed  of  by  Webber  within  68  short 
pages,  while  Wentworth's  Analytic  Geometry  covers  273  crowded  pages. 
Comparatively  much  space  was  given  by  Webber  to  the  applications  of 
mathematics,  such  as  gauging,  heights  and  distances,  surveying,  navi- 
gation, and  dialing.  These  practical  subjects  received  much  more  at- 
tention then  than  they  are  now  receiving  in  the  academic  department  in 
the  majority  of  our  colleges.  Webber  devotes  only  47  pages  to  the  im- 
portant and  extensive  subject  of  geometry,  and  gives  solutions  of  geo- 
metrical problems,  but  no  theorems.  This  apparent  neglect  of  the  oldest 
and  most  beautiful  of  mathematical  sciences  is  explained  by  Webber  in 
the  second  edition  of  his  work.  In  a  foot-note  (p.  339)  he  says  that  ''A 
tutor  teaches,  in  Harvard  College,  Playfair's  Elements  of  Geometry,  con- 
taining the  first  six  books  of  Euclid,  with  two  books  on  the  geometry  of 
solids.  Of  this  work  Mr.  F.  Nichols,  of  Philadelphia,  has  given  a  good 
American  edition  "  (1806).  Webber's  chapter  on  geometry  was,  there- 
fore,  intended  simply  as   a  book  with  problems  to  accompany  or 

*  "Old  Cambridge  and  New,"  by  Amory,  North  American  Review,  Vol.  114,  p.  118. 


INFLUX    OF    ENGLISH    MATHEMATICS.  61 

follow  Euclid.  If  the  course  in  elementary  geometry  was  taught  as  here 
indicated,  then  it  can  hardly  be  said  that  this  subject  was  neglected. 
Wherever  Euclid  is  diligently  studied,  there  geometry  is  not  slighted. 

Of  John  Farrar,  the  successor  of  Webber  in  the  chair  of  mathematics 
and  natural  philosophy,  we  shall  speak  when  we  consider  the  influx  of 
French  mathematics  into  the  United  States. 

Before  leaving  Harvard  College  we  shall  quote  two  short  passages 
taken  from  the  Harvard  Lyceum.  This  journal  was  a  publication  by  the 
students,  and  was  the  earliest  of  that  kind  at  this  college.  The  quota- 
tions about  to  be  given  disclose  an  effort  to  arouse  interest  among  stu- 
dents in  mathematical  studies.  In  the  first  number,  which  appeared 
July  14,  1810,  we  read  as  follows :  "  The  dry  field  of  mathematics  has 
brought  forth  most  ingenious  and  elegant  essays,  most  curious  and  eu- 
"tertaining  problems.  It  is  our  wish  to  construct  or  select  such  ques- 
tions in  their  various  branches  as  may  exercise  the  skill  of  our  corres- 
pondents in  their  solution."  This  promise  was  not  strictly  kept. 
Mathematical  enthusiasm  could  not  be  aroused  quite  so  easily.  There 
is  to  be  found,  to  be  sure,  an  ingenious  essay  on  mathematical  learning, 
presumably  written  by  a  Sophomore,  in  which  we  read:  "Perhaps  no 
science  has  been  so  universally  decried  by  the  overweeningly  dull  as  the. 
mathematics.  Superficial  dabblers  in  science,  contented  to  float  in 
doubts  and  chimeras,  and  unable  to  see  the  advantage  of  demonstra- 
ble truth,  turn  back  before  they  have  passed  the  narrow  path  which 
leads  to  the  firm  ground  of  mathematical  certainty,  and  not  willing  to 
have  others  more  successful  than  themselves,  like  the  Jewish  spies, 
they  endeavor  to  deter  them  from  the  way  by  horrid  stories  of  giant 
spectres  in  the  promised  land  of  demonstration,  and  scarcely  a  Caleb 
is  found  to  render  a  true  account  of  its  beauties."  But  the  Jewish  spies 
were  too  eloquent,  and  there  was  no  Caleb  to  furnish  curious  and  enter- 
taining problems,* 

YALE   COLLEGE. 

The  chair  of  mathematics  and  natural  philosophy  at  Yale  College 
was  established  in  1770.  Its  first  incumbent  was  Nehemiah  Strong, 
who  occupied  it  till  1781.  In  1794  Josias  Meigs  was  appointed  to  the 
position.  Meigs  was  graduated  at  Yale  in  1778,  and  served  as  tutor  in 
mathematics,  natural  philosophy,  and  astronomy,  from  1781  to  1784. 
.  In  1783  he  was  admitted  to  the  bar,  and  some  years  later  practiced  in 
-  Bermuda.  He  appeared  as  a  defender  of  American  vessels  that  were 
captured  by  British  privateers,  and  was,  in  consequence,  tried  for 
treason.  He  was  professor  at  Yale  until  1801,  teaching  mathematics, 
natural  philosophy,  and  chemistry.t    He  then  became  president  of  the 

'  The  Harvard  Book,  by  F.  O.  Vaille  and  H.  A.  Clark,  Vol.  II,  p.  174. 

tProf.  Beujamin  Silliraan  (class  of  1796)  says,  that  on  Kovember  4,1795,  "Mr. 
Meigs  beard  the  class  recite  at  noon,  as  Dr.  Dwigbt  is  owt  of  town.  Altbougb  Mr. 
Meigs  is  a  very  sensible  man  and  very  well  calculated  for  tbe  office  wbicb  be  now 
fills,  still  it  is  very  easy  to  make  a  contrast  between  him  and  the  president ;  but  I  am 


62  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

University  of  Georgia.  In  1812  be  was  appointed  surveyor-general,  and 
two  years  later,  Commissioner  of  the  General  Land  Office  of  the  United 
States. 

An  important  event  at  this  period  was  the  growth,  among  students, 
of  a  love  for  rhetoric  and  literature.  English  literature  had  hitherto 
been  quite  neglected.  A  taste  for  this  study  was  excited  by  two  young 
men,  John  Trumbull  and  Timothy  Dwight,  who  were  elected  tutors  in 
1771.  John  Trumbull  published,  daring  the  first  year  of  his  tutorship, 
a  poem  entitled  the  "  Progress  of  Dulaess,"  a  satire,  intended  to  expose 
the  absurdities  then  prevailing  in  the  system  of  college  instruction. 
Ancient  languages,  mathematics,  logic,  and  divinity  received,  in  his 
opinion,  an  altogether  disproportionate  amount  of  time.  In  his  poems, 
he  introduces  "  Dick  Hairbrain,"  a  town  fop,  ridiculous  in  dress  and 
empty  of  knowledge,  and  speaks  of  him  as  follows : 

"  What  tbougli  in  algebra,  his  station 

Was  negative  in  each  equation; 

Though  in  astronomy  survey'd, 

His  constant  course  was  retrograde ; 

O'er  Newton's  system  though  he  sleeps, 

And  finds  his  wits  in  dark  eclipse! 

His  talents  proved  of  highest  price  ' 

At  all  the  arts  of  card  and  dice  ; 

His  genius  turn'd  with  greatest  skill, 

To  whist,  loo,  cribbage,  and  quadrille, 

And  taught,  to  every  rival's  shame, 

Each  nice  distinction  of  the  game." 

Timothy  D wight's  love  for  literature  did  not  entirely  displace  his  in- 
terest for  mathematics.  On  the  contrary,  we  read  in  a  life  of  him  by 
his  son  that,  "In  addition  to  the  customary  mathematical  studies,  he 
carried  them  [his  students]  through  spherics  and  fluxions,  and  went  as 
far  as  any  of  them  would  accompany  him  into  the  Principia  of  New- 
ton." "  This,  however,  must  have  been  a  very  rare  thing,"  says  Presi- 
dent Woolsey.  Dwight  was  tutor  at  the  college  for  six  years.  To  ex- 
hibit his  continued  interest  in  mathematics  during  that  time  we  quote 
from  the  biography  of  him  the  following  passage :  "4^t  a  subsequent 
period,  during  his  residence  in  college  as  a  tutor,  he  engaged  deeply  in 
the  study  of  the  higher  branches  of  the  mathematics.  Among  the 
treatises  on  this  science  to  which  his  attention  was  directed,  was  l!^ew- 
ton's  Principia,  which  he  studied  with  the  utmost  care  and  attention, 
and  demonstrated,  in  course,  all  but  two  of  the  propositions  in  that 
profound  and  elaborate  work.  This  difficult  but  delightful  science,  in 
which  the  mind  is  always  guided  by  certainty  in  its  discovery  of  truth, 
so  fully  engrossed  his  attention  and  his  thoughts  that,  for  a  time,  he 

doubtful  whether  the  comparison  is  not  a  false  one,  because  the  president  is  one  of 
those  characters  which  we  very  seldom  meet  with  in  the  world,  and  who  form  its 
greatest  oruameuts."  (Barnard's  Educational  Journal,  vol.  26,  1S76,  International 
Series,  vol.  1,  p.  230.)  j 


INFLUX   OF   ENGLISH   MATHEMATICS.  63 

lost  even  his  relish  for  poetry ;  and  it  was  not  without  difficulty  that 
his  fondness  for  it  was  recovered." 

It  will  be  remembered  that  the  Mathematics  of  Ward  had  been  intro- 
duced before  the  Ee volution.  In  1788  Nicholas  Pike's  arithmetic  was 
adopted.  Soon  after  1801  Samuel  Webber's  Mathematics  displaced  the 
works  previously  used,  excepting  Euclid,  which  was  presumably  used 
during  this  whole  period  as  the  text-book  in  geometry.  At  about  the 
beginning  of  this  century  the  mathematical  course  was  as  follows: 
Freshmen,  Webber's  Mathematics ;  Sophomores,  Webber's  Mathematics 
and  Euclid's  Elements;  Juniors,  Enfield's  Natural  Philosophy  and  As- 
tronomy, and  Vince's  Fluxions ;  Seniors,  natural  philosophy  and  as- 
tronomy. 

A  strong  impetus  to  the  study  of  mathematics  in  American  colleges 
was  given  by  Jeremiah  Day.  He  graduated  from  Yale  in  1795,  became 
tutor  there  in  1798,  and  was  elected  i3rofessor  of  mathematics  and  nat- 
ural philosophy  in  1801.  Feeble  health  prevented  him  from  entering 
upon  the  duties  of  his  professorship  till  1803 ;  but  after  that  he  con- 
tinued in  them  till  1817,  when  he  succeeded  President  Dwight  in  the 
presidency  of  the  college.  The  chair  was  then  given  to  Alexander  Met- 
calf  Fisher. 

At  the  beginning  of  this  century  the  great  want  of  this  country  in 
the  department  of  pure  mathematics  was  /idequate  text-books.  Pro- 
fessor Webber,  of  Harvard,  was  the  first  who  attempted  to  supply  this 
want.  In  those  colleges  in  which  a  single  system  of  mathematics  had 
been  adopted,  preference  was  generally  given  to  the  "Mathematics  "of 
Webber.  But  his  compilation  was  rather  imperfectly  adapted  to  the 
purposes  for  which  it  was  made.  It  was  not  sufficiently  copious.  Many 
topics,  though  strictly  elementary  and  practically  important,  were 
passed  over  in  silence.  The  method  of  treatment  was  too  involved  and 
the  style  not  sufficiently  clear  to  make  the  subject  attractive  to  the 
young  student.  Accordingly  Professor  Day  set  himself  to  work  to 
write  a  series  of  books  which  should  supply  more  adequately  the  needs 
of  American  colleges.  In  1814  appeared  his  Algebra,  and  his  Mensura- 
tion of  Superficies  and  Solids,  in  1815  his  Plane  Trigonometry,  and  in 
1817  his  Navigation  and  Surveying.  It  was,  the  original  intention  of  the 
author  to  prepare  also  elementary  treatises  on  conic  sections,  spherics, 
and  fluxions,  but  on  his  elevation  to  the  presidency  he  abandoned  this 
design.  His  Algebra  passed  through  numerous  editions,  the  latest  ol 
which  was  issued  in  1852,  by  the  joint  labors  of  himself  and  Professor 
Stanley. 

Day's  books  are  very  elementary,  and  introduce  the  student  by  easy 
and  gradual  steps  to  the  first  principles  of  the  respective  branches. 
To  us  of  to  day,  they  appear  too  elementary  for  college  use,  but  it  must 
not  be  forgotten  that  at  the  time  they  were  prepared,  they  were  just 
what  was  needed  to  meet  the  demands  of  the  times.  Students  apply- 
ing for  matriculation  in  those  days  had  received  very  defective  prepara- 


64  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

tory  traioiug,  especially  in  matbematics.  With  such  un wrought  mate- 
rial before  him,  it  was  natural  for  the  teacher  to  show  his  preference 
for  a  text-book  in  which  every  process  of  development  and  reasoning 
was  worked  out  patiently  and  minutely  through  all  its  successive  steps. 
Day  took  for  a  model  the  diffuse  manner  of  Euler  and  Lacroix,  rather 
than  the  concise  and  abridged  mode  of  the  English  writers.  The  great 
danger  in  this  course  is  that  no  obstacles  are  left  to  be  removed  by  the 
student  through  his  own  exertion.  In  the  opinion  of  some  teachers, 
Day  has  laid  himself  open  to  criticism  by  carrying  the  principle  of  mak- 
ing mathematics  easy  somewhat  too  far.  It  is  no  little  praise  for  a  book 
written  at  that  time  to  say  that,  unlike  most  books  of  that  period, 
Day's  mathematics  did  not  encourage  the  cramming  of  rules  orthe  per- 
forming of  operations  blindly.  On  the  contrary,  the  diligent  student 
acquired  from  them  a  rational  understanding  of  the  subject. 

Day's  mathematics  were  at  once  everywhere  received  with  eager- 
ness. They  were  introduced  in  nearlj'^  all  our  colleges.  Even  at 
the  end  of  a  period  of  fifty  years  they  still  held  their  place  in  many 
of  our  schools.  In  view  of  these  facts,  "it  may  safely  be  said  that  the 
value  of  what  their  author  did  by  means  of  them  for  the  college  and  for 
the  country  at  large,  while  holding  the  office  of  professor  from  1803  to 
1817,  the  time  when  he  succeeded  Dr.  Dwight,  was  not  surpassed  by 
anything  in  science  and  literature  which  he  did  subsequently  during 
his  long  term  of  office  as  president  of  the  college."* 

As  a  teacher  and  writer,  President  Day  was  distinguished  for  the 
simplicity  and  clearness  of  his  methods  of  illustration.  His  kind- 
heartedness  and  urbanity  of  demeanor  secured  the  love  and  respect 
both  of  friends  and  pupils. 

He  was  succeeded  in  the  chair  of  mathematics  and  natural  philosophy 
by  Alexander  Metcalf  Fisher,  who  held  it  until  his  death  by  drowning 
in  1822,  at  the  shipwreck  of  the  Albion,  off  the  Irish  coast.  Fisher 
possessed  extraordinary  natural  aptitudes  for  learning.  He  had  pre- 
pared a  full  course  of  lectures  in  natural  philosophy,  both  theoretical 
and  experimental,  which  were  marked  for  their  copiousness  and  their 
exact  adaptation  to  the  purpose  of  instruction.  His  clear  conception 
of  what  a  text-book  should  be  is  well  shown  in  his  review  of  Enfield's 
Philosophy.t 

Regarding  the  course  in  natural  philosophy  at  Yale,  it  may  be  re- 
marked, that  in  1788  Martin's  Philosophy,  which  had  gone  out  of  print, 
was  succeeded  by  Enfield's  Natural  Philosophy,  first  published  in  1783. 
William  Enfield  was  a  prominent  English  dissenter.    He  preached  in 

*  Yale  College:  A  Sketch,  of  its  History,  by  William  Kingsley,  Vol.  I,  p.  115, 
t  American  Journal  of  Science,  Vol.  Ill,  1821,  p.  125.  lu  Vol.  V,  p.  83,  of  the  same 
journal,  is  an  article  by  him,  "On  Maxima  and  Minima  of  Functions  of  Two  Variable 
Quantities."  He  contributed  solutions  to  questions  in  the  American  Monthly  Maga- 
zine, and  in  Leybourn's  Mathematical  Repository  (London).  The  fourth  volume  of 
the  Memoirs  of  the  American  Academy  of  Arts  and  Sciences  contains  observations  by 
him  on  the  comet  of  1819  and  calculations  of  its  orbit. 


INFLUX   OF   ENGLISH   MATHEMATICS.  65 

TJnitariaii  churches  and  published  several  volumes  of  sermons.  Being 
engaged  chiefly  in  theological  studies,  comparatively  little  attention 
was  paid  by  him  to  the  exact  sciences.  Nevertheless,  he  succeeded  in 
compiling  a  work  on  natural  philosophy  which  possessed  elements  of 
popularity  and  was  used  in  our  American  colleges  for  four  decennia.  In 
1820  appeared  the  third  American  edition  of  this  work,  which  was 
then  used  by  nearly  all  the  seminaries  of  learning  in  New  England, 
notwithstanding  the  fact  that,  excepting  in  electricity  and  magnetism 
and  a  few  particulars  in  astronomy,  it  presented  hardly  any  idea  of 
the  progress  made  in  the  different  branches  of  philosophy  since  the 
period  of  Newton. 

UNIVERSITY  OF  PENNSYLYANIA. 

The  University  of  Pennsylvania,  which  had  such  a  remarkable  growth 
under  the  administration  of  Dr.  William  Smith,  before  the  Eevolation- 
ary  War,  had  a  comparatively  small  attendance  of  students  after  the 
war,  and  the  college  department  is  said  to  have  been  quite  inferior  to 
that  of  the  leading  American  colleges  of  that  time. 

An  educator  who  was  long  and  prominently  connected  with  this  in- 
stitution and  whose  activity  was  directed  towards  maintaining  and  rais- 
ing its  standard,  was  Robert  Patterson,  the  elder.  He  was  born  in 
1743  in  Ireland,  and  at  an  early  age  showed  a  fondness  for  mathematics. 
In  1768  he  emigrated  to  Philadelphia.  He  first  taught  school  in  Buck- 
ingham, and  one  of  his  first  scholars  was  Andrew  Ellicott,  who  after- 
ward became  celebrated  for  his  mathematical  knowledge  displayed  in 
the  service  of  the  United  States. 

About  this  time  Maskelyne,  the  astronomer  royal  of  England,  com- 
piled and  published  regularly  the  Nautical  Almana^c.  This  turned  the 
attention  of  the  principal  navigators  in  American  ports  to  the  calcula- 
tions of  longitude  from  lunar  observations,  in  which  they  were  eager  to 
obtain  instruction.  Patterson  removed  to  Philadelphia,  began  giving 
instruction  on  this  subject,  and  soon  had  for  his  scholars  the  most  dis- 
tinguished commanders  who  sailed  from  this  port.  Afteward  he  be- 
came principal  of  the  Wilmington  Academy,  Delaware,  and  in  1779 
was  appointed  professor  of  mathematics  and  natural  philosophy  at  the 
University  of  Pennsylvania,  which  post  he  filled  for  thirty-five  years. 
He  was  also  elected  vice-provost  of  that  institution.* 

Robert  Patterson  communicated  several  scientific  papers  to  the  Phil- 
osophical Transactions  (Vols.  II,  III,  and  IV),  and  was  a  frequent  con- 
tributor of  problems  and  solutions  to  mathematical  journals.  He  ed- 
ited James  Ferguson's  Lectures  on  Mechanics  (1806),  and  also  Fergu- 
son's "Astronomy  explained  upon  Sir  Isaac  Newton's  principles  and 
made  easy  to  those  who  have  not  studied  mathematics  "  (1809).    Fer- 

•  Transactions  of  American  Philosophical  Society,  Vol.  II,  New  Series,  Obituary 
Notice  of  Robert  Patterson,  LL.  D.,  late  President  of  the  American  Philosophical  So- 
ciety. 

881— No.  3 5 


66  TEACHma  and  history  of  mathematics. 

guson  was  a  celebrated  lecturer  on  astronomy  and.  mechanics  in  Eng- 
land, who  contributed  more  than  perhaps  any  other  man  there  to  the 
extension  of  physical  science  among  all  classes  of  society,  but  especially 
among  that  largest  class  whose  circumstances  preclude  them  from  a 
regular  course  of  scientific  instruction.  His  influence  was  strongly  felt 
even  in  this  country,  as  is  seen  from  the  American  editions  by  Eobert 
Patterson  of  his  astronomy  and  mechanics.  Patterson  wrote  a  small 
astronomy,  entitled  the  Newtonian  System,  which  was  published  in 
1808.  Ten  years  later  he  published  an  arithmetic,  elaborated  from  his 
own  written  compends,  previously  used  in  the  University.  Though 
lucid  and  ingenious,  this  arithmetic  was  rather  difficult  for  beginners, 
and  never  reached  an  extended  circulation. 

It  is  believed  by  many  that  mathematicians  generally  possess  a  strong 
memory  for  numbers.  This  was  certainly  not  true  of  Patterson,  for 
we  are  told  that  he  could  not  remember  even  the  number  of  his  own 
house.  He  met  this  dilemma  by  devising  a  mnemonic,  which  was  indeed 
worthy  of  a  mathematician.  The  number  of  his  residence  was  285, 
which  answered  to  the  following  conditions :  ''  The  second  digit  is  the 
cube  of  the  first,  and  the  third  the  mean  of  the  first  two."  It  is  to 
be  wondered  that,  during-  some  fit  of  intense  abstraction,  the  learned 
professor  did  not  pronounce  111  to  be  the  number  of  his  house,  instead 
of  285 ',  for  111  is  a  number  satisfying  the  above  conditions  quite  as 
well  as  285. 

When  Eobert  Patterson  resigned  his  position  at  the  University  of 
Pennsylvania  in  1814,  he  was  succeeded  by  his  son,  Eobert  M.  Patter- 
son. The  latter  was  graduated  at  the  University  in  1S04.  After  receiv- 
ing the  degree  of  M.  D,,  in  1808,  he  pursued  professional  studies  in 
Paris  and  London.  In  1814  he  was  appointed  professor  of  mathematics 
and  natural  philosophy,  which  office  he  filled  until  1828,  when  he  ac- 
cepted the  chair  of  natural  philosophy  at  the  University  of  Virginia. 
Eobert  M.  Patterson  published  no  mathematical  books. 

From  1828  to  1834  the  chair  of  mathematics  and  natural  philosophy 
was  filled  by  Prof.  Eobert  Adrain,  The  days  of  greatest  activity  of 
this  most  prominent  teacher  of  mathematics  were  spent  at  other  in- 
stitutions, but  we  take  this  opportunity  of  introducing  a  sketch  of  his 
life.* 

Eobert  Adrain  was  born  in  Ireland.  At  the  age  of  fifteen  he  lost  both 
his  parents,  and  thenceforward  he  supported  himself  by  teaching.  At 
the  end  of  an  old  arithmetic  he  found  the  signs  used  in  algebra.  His 
curiosity  becomiag  greatly  excited  to  discover  their  meaning,  he  gave 
himself  no  rest  until  at  last  he  found  out  what  they  meant.  In  a 
short  time  he  was  able  to  resolve  any  sum  in  the  arithmetic  by  algebra. 
Thenceforth  he  devoted  himself  with  enthusiastic  ardor  to  mathematics. 
He  took  part  in  the  Irish  rebellion  of  1798,  received  a  severe  wound,  and 

•  This  sketch  ia  extracted  from  an  article  in  the  Democratic  Eeview,  1844,  Vol, 
XIV. 


INFLUX    OF    ENGLISH    MATHEMATICS  67 

escaped  to  America.  Immediately  after  his  arrival  he  began  teaching 
in  New  Jersey.  After  two  or  three  years  he  became  principal  of  the 
York  County  Academy  in  Pennsylvania.  He  then  began  contributing 
problems  and  solutions  to  the  Mathematical  Correspondent,  a  journal 
published  in  New  York.  This  was  the  means  of  bringing  his  mathe- 
matical talents  before  the  public.  He  obtained  several  prize  medals, 
awarded  for  the  best  solutions. 

In  1805  he  moved  to  Eeading,  Pa.,  to  take  charge  of  the  academy  of 
that  place.  He  started  there  a  mathematical  journal  called  the  Analyst. 
The  first  number  was  published  in  Eeading,  but  its  typographical  exe- 
cution disappointed  him  so  much  that  he  employed  a  i)ublisher  at  Phila- 
delphia and  incurred  the  extra  expense  of  a  republication.  We  shall 
speak  of  this  journal  again  later. 

In  1810  he  was  called  to  the  professorship  of  mathematics  and  natural 
philosophy  at  Queen's  (now  Eutgers)  College;  and,  in  1813,  to  the  pro- 
fessorship of  mathematics  at  Columbia  College.  In  New  York  he  be- 
came the  center  of  attraction  to  those  pursuing  mathematical  studies. 
A  mathematical  club  was  established,  in  which  he  shone  as  the  great 
luminary  among  lesser  lights.  As  a  teacher,  he  had  a  most  happy 
faculty  of  imparting  instruction. 

In  1826  the  delicate  state  of  his  wife's  health  induced  him  to  leave 
Columbia  College  in  New  York  and  to  remove  to  the  pure  air  and 
healthful  breezes  of  the  country  near  New  Brunswick.  About  two 
years  later  he  was  induced  to  accept  the  professorship  of  mathematics 
at  the  University  of  Pennsylvania,  a  position  which  had  been  held  at 
the  beginning  of  the  century  by  the  well-known  Eobert  Patterson. 
Adrain  became  also  vice-provost  of  this  institution. 

He  resigned  this  position  in  1834  and  returned  to  his  country  seat 
near  New  Brunswick,  intending  to  pass  his  time  with  his  family  and 
in  study.  But  he  did  not  remain  there  long,  for  the  habit  of  teaching 
had  become  too  strong  easily  to  be  resisted.  He  moved  to  New  York 
and  taught  in  the  grammar  school  connected  with  Columbia  College 
until  within  three  years  of  his  death.  At  this  time  his  mental  faculties 
began  very  perceptibly  to  fail.  He  greatly  lamented  their  decay,  and, 
one  day  when  a  friend  called  in  to  see  him,  he  had  a  volume  of  La  Place 
on  his  lap  endeavoring  to  read  it.  "  Ah,"  said  he,  in  a  melancholy  tone 
of  voice,  "this  is  a  dead  language  to  me  now;  once  I  could  read  La 
~Place,  but  that  time  has  gone  by."    He  died  in  1843. 

Among  American  mathematicians  of  his  day,  Eobert  Adrain  was  ex- 
celled only  by  Nathaniel  Bowditch.  Of  his  many  contributions  to 
mathematical  journals,  one  of  the  earliest  was  an  essay  published  in 
1804  in  the  Mathematical  Correspondent  on  Diophantine  analysis.  This 
was  the  earliest  attempt  to  introduce  this  analysis  in  America. 

lu  1808  Adrain  began  editing  and  publishing  the  Analyst,  or  Mathe- 
matical Museum.  At  that  time  he  had  not  yet  entered  upon  his  career 
as  college  professor.    The  above  periodical  contained  chiefly  solutions 


68  TEACHING   AND    HISTOKY    OF   MATHEMATICS. 

to  mathematical  questions  proposed  by  the  various  contributors.  It  was 
a  small,  modest  publication,  which  had  only  a  very  limited  circulation  in 
this  country,  and  was  unknown  to  foreign  mathematicians.  It  lived, 
moreover,  only  a  very  short  time,  for  only  five  numbers  ever  appeared. 
And  yet,  this  apparently  insignificant  little  journal,  edited  by  a  teacher 
at  an  ordinary  academy,  contained  one  article  which  was  an  original 
contribution  of  great  value  to  mathematical  science.  It  was,  in  fact, 
the  first  original  work  of  any  importance  in  pure  mathematics  that  had 
been  done  in  the  United  States.  I  refer  to  Kobert  Adrain's  deduction 
of  the  Law  of  Probability  of  Error  in  Observation.  The  honor  of  the  first 
statement  in  printed  form  of  this  law,  com  monly  known  as  thei*rincipleof 
Least  Squares,  is  due  to  the  celebrated  French  mathematician  Legendre, 
who  proposed  it  in  1805  as  an  advantageous  method  of  adjusting  obser- 
vations. But  upon  Eobert  Adrain  falls  the  honor  of  being  the  first  to 
publish  a  demonstration  of  this  law.  He  does  not  use  the  term  "  least 
squares,"  and  seems  to  have  been  entirely  unacquainted  with  the 
writings  of  Legendre.  It  follows,  therefore,  that  not  only  the  two  de- 
ductions of  this  principle  given  by  Adrain  were  original  with  him,  but 
also  the  very  principle  itself. 

We  now  give  the  history  of  this  discovery  by  Adrain.  Eobert  Patter- 
son, of  the  University  of  Pennsylvania,  proposed  in  the  Analyst  the 
following  prize  question:  "In  order  to  find  the  content  of  apiece  Of 
ground,  *  *  *  Imeasured  with  a  common  circumferen tor  and  chain 
the  bearings  and  lengths  of  its  several  sides,  »  *  *  but  upon  cast- 
ing up  the  difference  of  the  latitude  and  departure,  I  discovered  *  *  * 
that  some  error  had  been  contracted  in  taking  the  dimensions.  E"ow,  it 
is  required  to  compute  the  area  of  this  inclosure  on  the  most  probable 
supposition  of  this  error."  This  was  proposed  in  No.  II  of  the  Analyst, 
and  after  being  a  second  time  renewed  as  a  prize  question  in  No.  Ill,  it 
was  at  length,  in  No.  lY,  solved  by  a  course  of  special  reasoning  by 
Nathaniel  Bowditch,  to  whom  Adrain  awarded  the  prize  of  ten  dollars. 
Immediately  following  Bowditch's  special  solution,  the  editor,  Adrain, 
added  his  own  solution  of  the  following  more  difficult  general  problem: 
«'  Eesearch  concerning  the  probabilities  of  the  errors  which  happen  in 
making  observations."  *  This  paper  is  of  great  historical  interest,  as 
containing  the  first  deduction  of  the  law  of  facility  of  error. 

cp  {x)  =  ce-^''' 

q)  [x]  being  the  probability  of  any  error  a?,  and  c  and  h  quantities  de- 

*  Aualj'st,  No.  rV,  pp.  93-97.  Copies  of  this  journal  are  very  rare.  No.  IV  is  to  be 
found  iu  the  Congressioual  Library  in  Washington ;  No.  Ill  and  No.  IV  are  in  the 
Library  of  the  American  Philosophical  Society,  Philadelphia.  Adrain's  first  proof  of 
the  Principle  of  Least  Squares  was  re-published  by  Cleveland  Abbe  in  the  American 
Journal  of  Science  and  Arts,  third  series,  1871,  pp.  411-415.  Adrain's  second  proof 
vcas  re-published  by  Mansiield  Merriman  in  the  Transactions  of  the  Connecticut  Acad- 
emy, Vol.  IV,  1887,  p.  164 ;  also  in  the  Analyst  (edited  and  published  by  J.  E.  Hen- 
dricks, Des  Moines,  lovra).  Vol.  IV,  No.  II,  p.  33. 


INFLUX    OF   ENGLISH   MATHEMATICS.  69 

pending  on  the  precision  of  the  measurement.  Adrain  gives  two  proofs 
of  this  law.  The  first  proof  depends  upon  the  "  self-evident  principle," 
as  he  calls  it,  that  the  true  errors  of  measured  quantities  are  propor- 
tional to  the  quantities  themselves.  The  arbitrary  nature  of  this  as- 
sumption is  pointed  out  by  J.  W.  L.  Glaisher  in  the  Memoirs  of  the 
Eoyal  Astronomical  Society,  Part  IT,  vol.  39,  1871-72.  "This,"  says 
Glaisher,  ''seems  very  far  from  being  evident,  not  to  say  very  far  from 
being  true,  generally.  One  would  expect  a  less  relative  error  in  a 
greater  distance."  Glaisher  raises  other  objections  to  Adrain's  first 
proof,  and  then  pronounces  it  entirely  inconclusive.  Adrain's  second 
proof,  which  is  essentially  the  same  as  that  given  later  by  John  Her- 
schel,  and  usually  called  Herschel's  proof,  is  likewise  defective,  as  has 
been  pointed  out  by  Prof.  Mansfield  Merriman. 

In  order  to  place  these  criticisms  on  Adrain's  two  demonstrations  in 
the  proper  light,  it  should  be  remarked  here  that  the  subject  of  which 
they  treat  is  one  of  great  difiSculty.  There  has  been  great  difierenee 
of  opinion  among  leading  mathematicians  as  to  what  assumptions  re- 
garding the  nature  of  errors  can  be  safely  and  legitimately  made,  and 
taken  as  a  basis  upon  which  to  construct  demonstrations  and  what  ones 
should  be  rejected  as  being  false  or  as  demanding  demonstration. 

Subsequently  to  Adrain's  paper,  proofs  were  published  hy  Gauss,  La 
Place,  Bessel,  Ivory,  John  Herschel,  Tait,  Donkin,  and  others.  Alto- 
gether, there  appeared  over  a  dozen  distinct  proofs,  but  all  of  these 
"  contain,  to  say  the  least,  some  point-  of  difficulty  "  (Glaisher).  If, 
therefore,  it  be  said  that  Adrain's  proofs  are  inconclusive,  we  must  re- 
member that  all  other  proofs  hitherto  given  possess  to  a  greater  or  less 
degree  the  same  defect. 

The  number  of  the  Analyst  which  gives  Adrain's  two  proofs  contains 
also  the  following  applications  of  this  method :  (1)  To  find  the  most 
probable  value  of  any  quantity  of  which  a  number  of  direct  measure- 
ments is  given ;  (2)  to  find  the  most  probable  position  of  a  point  in 
space;  (3)  to  correct  dead-reckoning  at  sea;  to  correct  the  bearings 
and  distances  of  a  field  survey. 

At  the  close  of  the  article  he  says:  "I  have  applied  the  principles 
of  this  essay  to  the  determination  of  the  most  probable  value  of  the 
earth's  ellipticity,  etc.,  but  want  of  room  will  not  permit  me  to  give  the 
investigation  at  this  time."  It  was  published  nine  years  later  in  Vol- 
ume I,  new  series,  of  the  Transactions  of  the  American  Philosophical 
Society  (papers  No.  IV  and  XXVII).  In  the  first  paper  he  finds  the 
earth's  ellipticity  to  be  gig  instead  of  -g-^e,  as  was  erroneously  given  by 
La  Place  (La  M6canique  Celeste,  Vol.  III).  In  the  second  paper  Adrain 
applies  his  rule  to  the  evaluation  of  the  mean  diameter  of  the  earth, 
^hich  he  finds  to  be  7,918.7. 

His  rule  for  correcting  dead-reckoning  at  sea  was  adopted  by  Dr. 
Bowditch  in  his  last  edition  of  his  Practical  Navigator.  Adrain's  rule 
for  correcting  a  survey  is  referred  to  by  John  Gum  mere  in  his  Survey- 


70  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

ing,  as  having  been  given  and  demonstrated  by  Bowditch  and  Adrain 
in  the  Analyst. 

It  thus  appears  that  these  rules  of  Adrain  were  made  use  of  by  at  least 
some  of  the  contemporary  American  mathematicians,  but  the  principles 
from  which  these  rules  were  deduced  and  the  demonstrations  of  these 
principles  appear  to  have  excited  little  attention,  and  to  have  been  soon 
forgotten.  Foreign  mathematicians  never  heard  of  Adrain's  investiga- 
tions on  the  subject  of  least  squares  until  Adrain's  first  proof  and  ex- 
tracts from  other  papers  were  reprinted  by  Cleveland  Abbe  in  the  Amer- 
ican Journal  of  Science  and  Arts  in  1871,  or  sixty  years  after  their  first 
publication  in  the  Analyst.  By  a  very  strange  oversight  Cleveland 
Abbe  does  not  even  mention  Adrain's  second  proof.  The  existence  of 
this  proof  was  pointed  out,  however,  a  few  years  later  by  Prof.  Mansfield 
Merriman. 

It  is  not  much  to  the  credit  of  American  mathematicians  that  they 
should  have  permitted  theoretical  investigations  of  such  great  value  to 
remain  so  long  in  obscurity.  Let  justice  be  done  to  Adrain,  and  let  him 
be  credited  "  with  the  independent  invention  and  a])plication  of  the 
most  valuable  arithmetical  x>rocess  that  has  been  invoked  to  aid  the 
progress  of  the  exact  sciences." 

By  the  numerous  elegant  solutions  which  Adrain  contributed  to  math- 
ematical journals  in  this  country,  by  his  labors  as  teacher  at  Eutgers 
College,  Columbia  College,  and  the  University  of  Pennsylvania,  by  his 
editions  of  Button's  Mathematics,  he  contributed  powerfully  to  the 
progress  of  mathematical  studies  in  the  United  States.  His  first  edition 
of  Hutton's  Mathematics  was  followed  in  1812  by  a  second  edition,  and 
in  1822  by  a  third  edition,  in  which  he  added  an  elementary  treatise  of 
sixty  pages  on  descriptive  geometry, "  in  which  the  principles  and  funda- 
mental problems  are  given  in  a  simple  and  easy  manner."  Other  edi- 
tions came  out  later.  Adrain's  edition  of  Hutton  contained  improve- 
ments in  method  and  important  corrections,  the  value  of  which  was  rec- 
ognized by  Mr.  Hutton  himself. 

It  may  be  well  to  call  to  mind  at  this  place  that  Charles  Hutton  was 
professor  of  mathematics  at  the  Royal  Military  Academy  at  Woolwich 
from  1773  to  1805.  His  course  of  mathematics  was  published  in  Lon- 
don, 1798-1801.  In  its  day  this  work  was  doubtless  the  best  of  its 
kind  in  the  English  language.  But  at  that  time  the  English  were  far 
behind  the  French  in  the  cultivation  and  teaching  of  mathematics. 
Hutton's  course  was  plain  and  simple,  but  defective  both  in  extent  and 
analysis.  The  English  works  of  that  day  generally  contained  rules 
without  principles,  and  were  decidedly  inferior  to  the  explanatory  trea- 
tises of  Lacroix  and  Bourdon,  then  used  in  France.  Hutton's  mathe- 
matics were  used  once  at  our  own  Military  Academy  at  West  Point, 
but  were  soon  exchanged  for  the  more  analytical  and  copious  treatises 
by  French  authors. 

We  close  our  remarks  on  Eobert  Adrain  with  the  following  quotation 


INFLUX   OF   ENGLISH   MATHEMATICS.  71 

from  the  Democratic  Eeview  of  1844,  Vol.  XIY :  "  He  published  little, 
because  he  was  too  severe  a  critic  upon  his  own  writings.  He  would 
revise  and  re-revise.  It  is  said  that  while  at  Columbia,  he  had  a  trea- 
tise  on  the  differential  and  integral  calculus  all  written  out  and  ready 
for  the  press ;  but  upon  giving  it  a  further  revision  he  became  dissat- 
isfied with  some  parts  of  it,  and  committed  the  whole  to  the  flames." 
He  left  a  number  of  manuscripts  with  commentaries  on  the  M6canique 
Analytique  of  Lagrange  and  the  M^canique  Celeste  of  La  Place.* 

COLLEGE  OF  NEW  JERSEY  (PRINCETON). 

The  College  of  New  Jersey  first  opened  at  Elizabethtown,  in  1746. 
Soon  after,  it  was  transferred  to  Newark,  and  in  1756  to  Princeton. 
About  seventy  students  moved  from  Newark  to  Princeton.  The  first  pres- 
ident died  after  having  been  in  office  less  than  a  year.  Hissuccessor, 
Aaron  Burr,  the  elder,  held  the  post  for  ten  years.  He  was  an  incessant 
worker  and  toiler.  Though  he  was  assisted  by  two  tutors,  he  was  him- 
self teacher  in  Greek,  logic,  ontology,  natural  philosophy,  and  in  the  cal- 
culation of  eclipses.]  The  courses  in  physics  were  illustrated  by  appa- 
ratus which  had  been  obtained  from  Philadelphia.  Popular  lectures 
were  delivered  on  the  new  electricity,  and  both  president  and  students 
repeated  Franklin's  experiment  on  the  influence  of  pressure  on  the  boil- 
ing point,  with  glass  globes  of  their  own. 

At  first  the  college  had  no  professors  with  fixed  functions  and  perma- 
nent salaries.  The  instruction  in  classics  and  mathematics  was  com- 
mitted to  tutors  who  had  lately  graduated  and  were  preparing  for  the 
ministry.    They  taught  generally  for  but  few  years. 

The  first  professor  of  mathematics  and  natural  philosophy  was  Will- 
iam Churchill  Houston.  In  early  manhood  he  entered  Princeton  College, 
taught  in  the  college  grammar  school,  and  was  graduated  in  1768.  He 
was  then  appointed  tutor,  and,  in  1771,  elected  professor.  At  the  be- 
ginning of  the  Revolutionary  War,  he  and  Dr.  Witherspoon  were  the 
only  professors  in  the  college.  When  Princeton  was  invaded  in  1776, 
and  the  college  was  closed,  he  took  active  part  in  the  war.  As  soon  as 
quiet  was  restored  at  Princeton,  he  resumed  his  college  duties.  Soon 
after  he  was  sent  as  a  representative  to  Congress.  He  resigned  his  chair 
in  1783.  In  midst  of  his  many  duties,  he  had  acquired  a  sufficient 
knowledge  of  law  to  be  admitted  to  the  bar.  As  a  lawyer  he  soon  ac- 
quired great  reputation. 

John  Adams  visited  Princeton  College  in  the  opening  days  of  the 
Revolution,  when  he  was  on  his  way  to  the  Continental  Congress.  In 
his  diary  of  August  26,  1774,  he  says :  "  Mr.  Euston,|  the  professor  of 
mathematics  and  natural  philosophy,  showed  us  the  library ;  it  is  not 
large,  but  has  some  good  books.    He  then  lead  us  into  the  apparatus 

*For  want  of  the  necessary  material,  our  sketch  of  the  University  of  Pennsylva- 
nia will  not  he  continued, 
+  The  College  Book,  by  Charles  F.  Eichardson  and  Henry  A.  Clark,  1878,  p.  97. 
X  Mr.  Houston  was  probably  intended.  t 


72  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

room ;  here  we  saw  a  most  beautiful  macbiue — an  orrery  or  planetarium, 
constructed  by  Mr.  Eittenhouse,  of  Philadelphia."  It  will  be  remembered 
that  both  the  University  of  Peunsylvania  and  Princeton  College  had 
been  negotiating  for  the  first  orrery  made  by  Kittenhouse,  and  that 
Princeton  carried  it  off,  much  to  the  chagrin  of  Dr.  William  Smith,  the 
president  of  the  University  of  Pennsylvania. 

The  chair  of  mathematics  and  natural  philosophy  was  filled  two  years 
after  the  resignation  of  Houston  by  the  appointment  of  Ashbel  Green. 
He  was  a  native  of  New  Jersey,  and  was  graduated  at  Princeton  Col- 
lege in  1783.  He  entered  the  ministry,  and  was  professor  for  the  two 
years  succeeding  1785.    Later,  he  became  president  of  the  institution. 

Green's  successor  was  Dr.  Walter  Minto,  a  Scotch  mathematician 
of  eminence.  He  was  graduated  at  the  University  of  Edinburgh,  and 
then  became  tutor  to  the  sons  of  George  Johnstone,  a  member  of  Parlia- 
ment. With  them  he  travelled  over  much  of  Europe.  In  Pisa  he  be- 
came acquainted  with  Dr.  Slop,  the  astronomer,  and  through  him  with 
the  then  novel  application  of  the  higher  analysis  to  planetary  motion. 
After  returning  to  Scotland  he  became  teacher  of  mathematics  at 
Edinburgh.  He  came  to  the  United  States  in  1786,  and  one  year  after 
became  connected  with  Princeton  College.  Before  coming  to  America  he 
published  a  Demonstration  of  the  Path  of  the  New  Planet;  Researches 
into  Some  Parts  of  the  Theory  of  Planets;  and  (with  Lord  Buchan)  an 
Account  of  the  Life,  Writings,  and  Inventions  of  Napier  of  Merchiston. 

While  at  Princeton,  he  delivered  on  the  evening  preceding  the  annual 
commencement  of  the  year  1788  "an  inaugural  orationonthe  Progress 
and  Importance  of  the  Mathematical  Sciences."  In  this  address  he 
traces  the  history  of  mathematics  down  to  the  time  of  Newton,  then 
directs  his  remarks  to  the  students  and  trustees,  emphasizing  the  im- 
portance of  mathematical  study.  "  The  genius  of  Newton,"  says  he, 
"  had  he  been  born  among  the  Indians,  instead  of  discovering  the  laws  of 
the  universe,  would  have  been  limited  to  the  improvement  of  the  in- 
struments of  hunting,  or  to  the  construction  of  commodious  wigwams." 
At  the  time  when  this  address  was  delivered  he  had  been  professor  at 
Princeton  about  a  year.  Near  the  close  of  his  oration  he  says :  "  It  gives 
me  a  deal  of  pleasure,  gentlemen,  to  have  occasion  to  observe,  in  this 
public  manner,  tha^t  a  considerable  majority  of  those  of  you  who  have 
studied  the  mathematics  under  my  direction  have  acquitted  yourselves 
even  better  than  my  expectations,  which,  believe  me,  were  very  san- 
guine." This  inaugural  address  is  his  only  publication  while  he  was 
connected  with  Princeton  College,  but  the  college  library  contains  some 
careful  and  curious  MSS.  on  mathematical  analysis  written  by  him. 
Doctor  Minto  died  at  Princeton  in  1796. 

The  mathematical  duties  were  now  assigned  to  Dr.  John  Maclean,  a 
native  of  Scotland.  In  his  day  he  was  one  of  the  most  distinguished 
professors  at  Princeton,  "the  soul  of  the  faculty."  His  specialty  was 
chemistry,  which  he  had  studied  in  Paris.    He  is  said  to  have  been  one 


INFLUX    OF   ENGLISH   MATHEMATICS.  73 

of  the  first  to  reproduce  in  America  the  views  of  the  new  French  school 
in  chemistry.  Daring  seven  years  he  was  professor  not  only  of  chem- 
istry, but  also  of  natural  history,  mathematics,  and  natural  philosophy; 
and  after  a  short  interval  of  four  years,  during  which  he  was  relieved 
from  mathematical  instruction  by  the  appointment  of  Dr.  Andrew 
Hunter  to  those  duties,  he  again  assumed  charge  of  all  the  scientific 
instruction  given  to  the  students.     He  died  in  1814. 

From  1812  to  1817,  Elijah  Slack,  a  graduate  of  Princeton  and  a  min- 
ister, was  professor  of  natural  philosophy  and  chemistry.  He  taught 
also  mathematics.  He  was  afterwards  president  of  Cincinnati  College. 
Henry  Yethake  taught  mathematics  from  1817  to  1821.  In  1823,  Mr. 
John  Maclean,  a  young  man  of  only  twenty-three  years,  was  made  pro- 
fessor of  mathematics. 

It  may  here  be  remarked  that  in  the  library  of  Princeton  College  there 
is  a  folio  volume  of  great  interest  and  value,  containing  a  copy  of  the 
first  printed  edition  of  Euclid's  Elements  in  Greek  (Basle,  1533);  the 
commentary  of  Proclus  on  the  First  Book  of  Euclid  (Basle,  1533) ;  a 
twofold  Latin  translation  (Basle,  1558) — one  the  Adolard-Oampanus 
version,  from  the  Arabic,  the  other  the  first  translation  into  Latin  from 
the  Greek  from  Theon's  Revision.  This  folio  was  once  the  property  of 
Henry  Billingsley,  who  three  hundred  years  ago  made  the  first  trans- 
lation of  Euclid  into  English.  By  the  examination  of  this  folio.  Dr.  G. 
B.  Halsted  was  able  to  show  that  the  first  English  translation  was  made 
from  the  Greek,  and  not,  as  was  formerly  supposed,  from  any  of  the 
Arabic-Latin  versions.* 

DARTMOUTH  COLLEGE. 

Dartmouth  College,  at  Hanover,  was  chartered  in  1769.  Wheelock 
was  the  first  president,  and  his  first  associate  in  instruction  as  tutor 
was  Bezaleel  Woodward,  who  had  graduated  at  Tale  in  1764,  during 
the  presidency  of  Clap,  of  whom  it  was  said  that  in  mathematics  and 
natural  philosophy  '*  he  was  not  equalled  by  more  than  one  man  in 
A  merica." 

Three  of  Dartmouth's  first  class  were  prepared  for  college  at  the 
"Indian  Charity  School"  in  Lebanon,  and  passed  their  first  three  years 
at  Yale. 

The  facilities  for  acquiring  a  classical  and  scientific  education  appear 
to  have  been  substantially  the  same  at  Dartmouth,  at  the  outset,  as  in 
other  American  colleges  of  that  period.f  Some  notion  regarding  the 
mathematical  course  at  thi's  college  may  be  drawn  from  a  letter  written 
in  1770  by  Nathan  Teasdale,  a  learned  and  indefatigable  teacher  in 
eastern  Connecticut,  to  Dr.  Wheelock,  the  president  of  the  college, 
introducing  one  of  Teasdale's  pupils,  who  applied  for  admission  to  the 
Senior  year.    The  young  man  is  described  as  "a  genius  somewhat 

*Note  on  the  First  Englisli  Enclid,  American  Journal  of  Mathematics,  Vol.  II,  1879. 
t  History  of  Dartmouth  College,  by  B.  P.  Smith,  p.  58. 


74  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

better  than  common,"  who  ''  had  made  excellent  progress.'^  "In  arith- 
metic, vulgar  and  decimal,  he  is  well  versed.  I  have  likewise  taught 
him  trigonometry,  altimetry,  longimetry,  navigation,  surveying,  dial- 
ing, and  gauging."  "  He  likewise  studied  Whiston's  Astronomy,  all 
except  the;  calculations."  We  are,  probably,  not  far  from  the  truth,  if 
we  conclude  that  the  studies  here  enumerated  were,  in  substance,  the 
mathematics  then  studied  at  Dartmouth  during  the  first  three  years. 

The  first  twelve  or  thirteen  years  were  years  of  very  great  trial  for 
Dartmouth.  The  funds  of  the  college  were  small  and  the  students  few. 
The  Revolutionary  War,  though  it  did  not  interrupt  the  college  exercises 
and  disperse  the  students,  must  have  diminished  their  number  and 
affected  their  spirits.  As  in  other  localities,  so  in  New  Hampshire,  the 
means  of  fitting  for  college  were  very  imperfect  and  many  of  the  college 
studies  were  inadequately  pursued.  "  I  remember,"  says  Samuel  Gil- 
man  Brown,*  "  hearing  one  of  the  older  graduates  say  that  the  first 
lesson  of  his  class  in  mathematics  was  twenty  pages  in  Euclid,  the 
instructor  remarking  that  he  should  require  only  the  captious  of  the 
propositions,  but  if  any  doubted  the  truth  of  them  he  might  read  the 
demonstrations,  though  for  his  part  his  mind  was  perfectly  satisfied." 
In  stories  like  this,  however,  we  must  allow  something  for  the  genius 
of  the  narrator.  This  story,  if  not  true,  is  certainly  of  the  ben  trovato 
sort.  The  requirements  for  admission  to  American  colleges  in  those 
days  were  low,  and  the  system  of  choosing  the  tutors,  to  whose  care 
the  Freshmen  and  Sophomore  classes  were  entirely  committed,  was 
enough  to  destroy  any  chances  of  rectifying  the  errors  of  bad  and 
insufficient  preparation.  Not  unfrequently  a  fresh  graduate  who  had 
excelled  in  classics  alone,  with  very  little  taste  for  mathematics,  would 
be  chosen  to  fill  a  tutorial  vacancy  requiring  him  to  teach  mathe- 
matics, and  vice  versa.  The  bad  consequences  of  such  a  system  need 
not  be  dwelt  upon  here. 

We  see  from  the  above  that  Euclid's  geometry  had  been  introduced 
in  the  early  days  of  the  college. 

In  1790  the  studies  in  college  were  as  follows  :  t 

"The  Freshmen  study  the  learned  languages,  the  rules  of  speaking, 
and  the  elements  of  mathematics.  The  Sophomores  attend  to  the  lan- 
guages, geography,  logic,  and  mathematics.  The  Junior  Sophisters, 
beside  the  languages,  enter  on  natural  and  moral  philosophy  and  com- 
position. The  Senior  class  compose  in  English  and  Latin  j  study  meta- 
physics and  the  elements  of  natural  and  political  law. 

"  The  books  used  by  the  students  are  Lowth's  English  Grammar, 
Perry's  Dictionary,  Pikers  Arithmetic,  Guthrie's  Geography,  Ward^s 
Mathematics,  Atkinson's  Epitome,  EammoticPs  Algebra,  Martinis  and 
EnfieWs  Natural  Philosophy,   Ferguson's   Astronomy,    Locke's   Essay, 

'Address  before  tbe  Society  of  Alumni  of  Dartmouth  College,  1855,  p,  17. 
t  Barnard's  Journal  of  Education,  Vol.  26,  1876,  lutemational  Series,  Vol.  I,  p.  278, 
quoted  by  Judge  Crosby  from  Belknap's  History  of  New  HampsMre,  p.  296. 


INFLUX  OF  ENGLISH   MATHEMATICS.  75 

Montesquieu's  Spirit  of  Laws,  and  Burlamaqui's  Natural  and  Political 
Law."  Hammond's  Algebra  was,  we  believe,  an  English  work.  In  a 
catalogue  of  old,  second-hand  books  we  find,  "  Hammond,  N.,  Elements 
of  Algebra  in  a  new  and  easy  method,  etc.,  8vo.  calf,  1742." 

Of  the  early  graduates  of  Dartmoutli  we  would  mention  Daniel 
Adams,  of  the  class  of  1797,  who  furnished  the  schoolboy's  satchel  with, 
the  Scholar's  Arithmetic,  one  of  the  best  and  most  popular  books  of  the 
time. 

Another  graduate  somewhat  distinguished  in  the  mathematical  line 
was  John  Hubbard,  of  the  class  of  1785.  After  studying  theology,  he 
became  preceptor  of  the  New  Ipswich  and  Deerfleld  academies  in  Mas- 
sacliusetts.  Afterwards  he  was  judge  of  probate  of  Cheshire  County, 
N.  H.  In  1804  he  succeeded  B.  Woodward  in  the  chair  of  mathematics 
and  natural  philosophy  at  Dartmouth,  and  filled  it  till  his  death  in  1810. 
He  published  an  Oration,  Rudiments  of  Geography,  The  American 
Eeader,  and  an  essay  on  Music,  but  nothing  on  mathematics. 

For  twenty-three  years,  beginning  in  1810,  Ebenezer  Adams  was  pro- 
fessor of  mathematics  and  natural  philosophy.  In  1833  he  was  made 
professor  emeritus. 

For  1824  the  mathematical  studies  as  indicated  in  the  catalogue  of 
the  college,  was  as  follows :  The  Freshmen  reviewed  "  arithmetick  "  and 
then  studied  algebra  during  the  third  term.  No  mathematical  studies 
are  given  for  the  first  two  terras.  The  Sophomores  were  put  down  for 
six  books  of  Euclid  during  the  first  term,  i^lane  trigonometry  and  its 
usual  applications  during  the  second  term,  and  the  completion  of  Euclid 
during  the  third  term.  The  Juniors  studied  "  conick  "  sections,  and 
"  spherick"  geometry  and  trigonometry  during  the  first  term.  The  rest 
of  the  year  was  given  to  natural  philosophy  and  astronomy.  No  mathe- 
matics in  the  last  year. 

In  1828  the  college  course  was  the  same,  but  algebra  to  the  end  of 
simple  equations  .was  added  to  the  terms  for  admission. 

BOWDOIN  COLLEGE.* 

When  Bowdoin  College  was  first  organized,  in  1802,  the  requirements 
for  admission  were  an  acquaintance  with  the  "fundamental  rules  of 
arithmetic."  Later,  the  expression  "  well- versed  in  arithmetic  "  is  used. 
The  first  definite  increase  in  the  requirements  did  not  occur  till  1834, 
when  part  of  algebra  was  added. 

During  the  first  three  years  of  its  existence  the  college  had  no  regular 
professor  of  mathematics.  But  in  1805  the  faculty  was  reinforced  by 
the  arrival  of  Parker  Cleaveland,  who  six  years  before  had  graduated 
at  Harvard  first  in  his  class  and  had  been  tutor  in  the  university.  The 
department  of  mathematics  and  natural  philosophy  was  assigned  to  the 

*  For  t^e  greater  part  of  the  material  used  in  writing  tliis  sketch,  the  writer  is  in- 
debted to  Prof.  George  T.  Little  of  Bowdoin  College. 


76  TEACHING    AND    HISTORY    OF    aIATHEMATICS. 

youthful  instructor.  He  remained  at  Bowdoin  till  his  death  in  1868 
and  earned  for  himself  the  enviable  reputation  of  "  Father  of  American 
mineralogy."  Cleaveland  was  professor  of  mathematics  from  1805  till 
1835.  One  of  the  first  books  used  was  Michael  Walsh's  arithmetic, 
published  at  Newburyport  in  1801.  Webber's  mathematics  were  taught 
for  many  years,  until  they  were  displaced  by  Farrar's  "  Cambridge 
mathematics." 

The  course  in  mathematics  at  the  beginning  and  for  twenty  years 
after,  was,  in  the  Freshman  year,  arithmetic ;  Sophomore  year,  algebra, 
geometry,  plane  trigonometry,  mensuration  of  surfaces  and  solids ;  Jun- 
ior year,  heights  and  distances,  surveying,  navigation,  conic  sections ; 
Senior  year,  spherical  geometry  and  trigonometry  with  application  to 
astronomy.  Algebra  was  gradually  forced  back  to  the  Freshman  year, 
but  a  part  of  the  first  term  of  this  year  was  given  to  arithmetic  as  late 
as  1850. 

In  regard  to  the  instruction  in  mathematics  during  the  professorship 
of  Parker  Cleaveland,  Professor  Little  sends  us  a  copy  of  a  letter  from 
their  oldest  living  graduate,  the  Kev.  Dr.  T.  T.  Stone,  of  the  class  of 
1820.  Says  he:  "Until  near  the  close  of  our  college  life  we  had  but 
one  professor  with  the  president  and  two  tutors.  Professor  Cleaveland 
added  to  his  duties  as  teacher  of  the  natural  sciences,  in  particular 
chemistry,  mineralogy,  and  such  as  were  contained  in  Enfield's  Natural 
Philosophy,  those  of  instructor  in  mathematics;  although,  I  think,  in 
the  latter,  that  is,  in  mathematics,  one  of  the  tutors  took  part.  Of  the 
tutors  who  had  most  to  do  with  this  department,  I  remember  Joseph 
Huntington  Jones,  afterward  a  Presbyterian  minister  in  Philadelphia; 
Samuel  Greene,  well  known  in  later  days  as  minister  of^I  think,  the 
Essex  Street  Church,  Boston;  and  Asa  Cummings,  minister  soon  after 
of  the  First  Church  in  North  Yarmouth  (the  North  since  dropped  off), 
and,  later  still,  editor  for  many  years  of  the  Christian  Mirror,  and,  if  I 
am  not  mistaken,  other  tutors  sometimes  assisted  in  the  department,  as 
Mr.  Newman,  who  from  tutor  became  professor  of  the  ancient  languages 
in  the  spring  of  1820,  and  afterward  professor  of  rhetoric.  It  was  he, 
unless  my  memory  fails  me,  who  took  our  class  out  to  survey  a  piece  of 
land  to  the  north  or  south-west  of  what  was  then  the  college  grounds, 
including  probably  the  place  where  he  and  Professor  Smyth  and  Pro- 
fessor Packard  afterward  lived— the  only  thing  connected  with  the 
mathematics  which  I  now  remember  outside  of  the  recitations  and  the 
preparation,  such  as  they  were,  for  the  regular  exercises. 

"  Of  the  books  then  used,  the  first,  and  that  which  went  with  us,  I 
am  not  sure  but  through  the  whole  course  from  the  Freshman  year  to 
the  Senior,  was  Webber's  Mathematics.  The  only  other  book  of  pure 
mathematics  was  Playfair's  edition  of  Euclid,  in  which  we  went  through 
so  much,  I  now  forget  how  much,  as  we  had  time  of  the  six  books  which 
comprised  a  large  part  of  the  work.  Added  to  this,  Enfield's  Philoso- 
phy took  in,  with  its  natural  science,  not  a  little  of  mathematical  illus- 
tration. 


INFLUX    OF   ENGLISH    MATHEMATICS.  77 

"  Of  the  methods  of  instructioD,  I  have  already  stated  that  the  only 
exception  I  remember  to  simple  recitation  was  a  single  slight  piece  of 
surveying.  We  were  required  to  study  the  prescribed  lesson  in  the 
book,  then  to  repeat  it,  not  of  course  word  for  word,  but  distinctly,  to 
the  professor  or  tutor  at  the  recitation  j  that  was  all." 

GEOEGETOWN  COLLEGE.* 

Shortly  after  the  close  of  the  American  Revolution  the  idea  of  es- 
tablishing a  college  in  Maryland,  then  the  chief  seat  of  the  Catholic 
religion  in  this  country,  presented  itself  to  the  Eev.  John  Carroll, 
afterwards  first  Archbishop  of  Baltimore.  Buildings  were  erected  for 
this  purpose  in  1789,  and  a  school  first  opened  two  years  later.  It 
rapidly  grew  into  favor.  Great  attention  was  then  paid  to  the  classic 
languages,  but  only  little  to  mathematics.  Until  1806,  when  the  col- 
lege came  into  the  hands  of  the  Jesuits,  the  school  was  rather  of 
preparatory  grade.    At  this  time  a  regular  college  course  was  arranged. 

In  1807  Fr.  James  Wallace  came  to  Georgetown.  He  had  the  classes 
here  for  two  years.  He  was  then  sent  to  New  York,  where  he  taught  in 
the  "IsTew  York  Literary  Institution,"  an  offshoot  of  Georgetown.  While 
in  New  York  he  published  a  work  on  the  Use  of  the  Globes  (New  York, 
1812 ).  He  returned  to  Georgetown  in  1813  or  1814,  and  remained  there 
until  1818,  when  he  removed  to  Charleston,  S.  0.  In  1821  his  connec- 
tion with  the  Society  of  Jesus  was  severed.  After  leaving  Georgetown, 
he  was  for  several  years  i^rofessor  of  mathematics  in  the  South  Caro- 
lina College.  During  his  second  stay  at  Georgetown  he  solved  a  prob- 
lem proposed  by  the  French  Academy ;  as  a  reward  they  sent  him 
many  fine  mathematical  works.  Professor  Wallace  was  a  man  of  ability, 
and  a  most  jjatient  and  successful  teacher. 

Eev.  Thomas  C.  Levins,  born  March  15, 1791,  taught  here  from  1822 
till  1825.  He  studied  at  Edinburgh,  under  Leslie,  and  then  taught  at 
Stonyhurst  College,  England.  In  1825  he  went  to  New  York.  Dr. 
Shea,  in  his  Catholic  Church  in  the  United  States  (p.  403),  states  that 
Fr.  Levins  was  one  of  the  engineers  of  the  Croton  Aqueduct.  He  died  in 
New  York,  May  6,  1843. 

We  have  not  been  able  to  obtain  more  definite  information  on  the 
early  mathematical  teaching  at  this  college. 

UNIYERSITY    OF  NORTH  CAROLINA. 

The  first  impulse  towards  the  establishment  of  a  university  in  North 
Carolina  came,  it  seems,  from  the  Scotch-Irish  element  occupying  the 
midland  belt  of  the  State.  '^  The  early  emigrants  and  settlers  of  this 
people  brought  their  preachers,  who  also  filled  the  office  of  teachers  for 
the  young.  Tradition  informs  us  that  the  most  popular  and  best  sus- 
tained of  these  nurseries  of  the  young  were  located  in  the  influential 

*  The  above  information  is  drawn  mainly  from  a  letter  of  Prof.  J,  F.  Dawson,  S. 
J.  ^  professor  of  physics  and  mechanics  at  Georgetown  College. 


78  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

counties  of  Iredell,  Mecklenburgh,  Guilford,  and  Orange.  It  was  from 
these  nurseries  came  the  desire  for  higher  education  that  formulated 
the  article  that  decreed  a  State  university.  Doubtless  the  granting  of 
a  charter  for  William  and  Mary  and  for  Harvard  by  the  royal  preroga- 
tive of  the  mother  country,  and  the  refusal  of  a  like  charter  to  Queen's 
College  at  Charlotte,  in  Mecklenburgh,  during  the  colonial  government, 
angered  the  Jiornets,  fired  the  resentment  of  the  Eevolutionary  pa- 
triots, and  quickened  their  action  in  the  blessings  of  liberty  under  the 
shield  of  the  new-born  Eepublic."* 

The  doors  of  the  university  first  opened  for  the  admission  of  students 
in  1795.  It  was  organized  after  the  model  of  Princeton  College,  which, 
in  turn,  was  patterned  after  the  Scottish  universities.  Shortly  after  the 
University  of  North  Carolina  had  begun,  Charles  W.  Harris,  a  graduate 
of  Princeton  College,  was  elected  to  the  professorship  of  mathematics. 
He  occupied  this  chair  for  only  one  year.  It  had  been  his  original  pur- 
pose to  study  law,  and  after  one  year's  experience  in  teaching  he  re- 
signed in  order  to  enter  the  legal  profession.  He  was  regarded  as  a 
man  of  considerable  ability,  but  he  died  at  the  age  of  33. 

He  was  succeeded  by  Rev.  Joseph  Caldwell,  who  was  also  a  graduate 
of  Princeton  and  a  native  of  ISew  Jersey.  He  had  been  one  year  tutor 
at  his  alma  mater.  This  remarkable  man  gave  for  nearly  forty  years 
his  best  energy  to  the  interests  of  the  university.  In  1804  he  was 
elected  president,  which  office  he  retained  till  his  death  in  1835,  with 
the  exception  of  four  years,  from  1812  to  1816,  during  which  period  he  re- 
tired voluntarily  to  the  professorship  of  mathematics  so  as  to  secure 
more  time  for  the  study  of  theology. 

At  first  the  faculty  was  very  small.  In  1814  it  consisted  of  "  President 
Caldwell,  Professor  Bingham,  and  Tutor  Henderson.  Their  college 
titles  were  'Old  Joe,'  'Old  Slick,'  and  'Little  Dick.'  '  Old  Joe,'  how- 
ever, was  only  thirty  years  of  age,  and  possessed  *  *  *  a  formi- 
dable share  of  youthful  activity ."t 

It  is  not  generally  known  that  Dr.  Caldwell,  in  August,  1832,  com- 
pleted the  first  college  observatory  built  in  the  United  States.  "  It 
was,"  says-  Professor  Love,  "  a  brick  structure  about  25  feet  high,  and 
contained  a  transit,  an  altitude  and  azimuth  instrument,  a  portable 
telescope,  an  astronomical  clock  with  mercurial  pendulum,  and  other 
minor  apparatus,  all  of  which  he  bought  in  London  in  1824  from  the 
best  makers.  For  want  of  means  and  interest,  however,  the  observa- 
tory, after  Dr.  Caldwell's  death  in  1835,  was  permitted  to  go  down,  and 

•Address  by  Paul  C.  Cameron  ia  the  inaugural  proceedings  at  the  University  of 
North  Carolina,  June  3,  1885,  p.  9.  All  the  material  for  this  sketch  of  mathematical 
teaching  at  that  university  has  been  furnished  to  the  writer  by  Prof.  James  Lee 
Love,  associate  professor  of  mathematics  at  the  University  of  North  Carolina.  Prof. 
Love  has  not  only  forwarded  pamphlets,  hut  has  himselt  made  careful  investigation 
into  the  history  of  the  institution,  and  kindly  communicated  his  results  to  the 
writer. 

+  Fifty  Years  Since,  by  William  Hooper,  1859,  p.  10. 


INFLUX   OP  ENGLISH  MATHEMATICS.  79 

even  the  records  of  observations  made  there  from  1833  to  1835  are  not 
now  known  to  exist."* 

Professor  Caldwell  was  a  man  of  liberal  and  progressive  views.  He 
laid  wisely  the  foundations  of  a  great  university  in  library  and  philo- 
sophical apparatus,  as  well  as  in  the  courses  of  study  and  in  the  men  he 
gathered  around  him  in  his  faculty.  In  remembrance  of  his  long  and 
untiring  devotion  to  the  iostitution  a  monument  has  been  erected  to 
him,  by  the  alumni,  in  a  grove  surrounding  the  university. 

When  Caldwell  went  to  Chapel  Hill  he  found  the  college  in  a  feeble 
state,  nearly  destitute  of  buildings,  library,  and  apparatus ;  the  stu- 
dents were  very  rough.  We  read  of  "unpleasant  upheavals  and  vol- 
canic eruptions  "  among  them.  Moreover,  the  bill  of  fare  with  which 
the  minds  of  the  students  were  obliged  to  content  themselves  was  very 
meager.  For  admission  in  mathematics  the  elements  of  arithmetic 
were  required  from  the  beginning,  in  1795,  to  1835.  In  1800  the  require- 
ment was  "  arithmetic  as  far  as  the  rule  of  three;"  in  1834,  "arithmetic 
to  square  root."  In  our  early  arithmetics  the  rule  of  three  was  given 
for  integers  before  fractions  were  touched  upon,  and  we  imagine  that 
fractions  were  not  required  for  admission,  nor  even  any  knowledge  of 
integral  arithmetic  beyond  the  merest  elements.  The  mathematical 
course  offered  in  1795  was  as  follows  :  (1)  Arithmetic  in  a  scientific  man- 
ner; (2)  algebra,  and  the  application  of  algebra  to  geometry;  (3)  Uu- 
elides  elements  ;  (4)  trigonometry  and  its  application  to  mensuration  of 
heights  and  distances,  of  surfaces  and  solids,  and  to  surveying  and  navi- 
gation ;  (5)  Conie  sections  ;  (6)  the  doctrine  of  the  sphere  and  cylinder  ; 
(7)  the  projection  of  the  sphere  ;  {8)  spherical  trigonometry;  (9)  the  doc- 
trine of  fluxions  ;  (10)  the  doctrine  of  chances  and  annuities.  "The  first 
four  courses,"  says  Professor  Love,  "were  to  be  required  for  graduation. 
The  remaining  courses  were  to  be  taught  if  requested,  but  they  were  not 
requested.''^ 

The  text-books  used  prior  to  1868  cannot  now  be  entirely  deter- 
mined. The  first  algebra  used  was  probably  Thomas  Simpson's.  It 
was  certainly  studied  in  1803  and  in  1815,  and,  perhaps,  as  late  as  1826. 
The  first  geometry  studied  was  Eobert  Simson's  Euclid.  On  the  ap- 
plication of  trigonometry  to  mensuration,  Ewing's  Synopsis  was  used 
first— certainly  as  early  as  1798.  About  1810  President  Caldwell  pre- 
pared a  course  in  geometry,  based  on  Simson's  Euclid.  ,This  was  used  by 
the  students  in  manuscript,  copies  having  been  made  and  handed  down 
from  class  to  class.  Hutton's  Geometry  was  introduced  in  1816.  In  1822 
Dr.  Caldwell  published  his  geometry,  under  the  title,  "A  Compendious 
System  of  Elementary  Geometry."  It  was  used  for  some  years.  Bound 
with  this  book  in  one  volume  was  a  treatise  on  trigonometry.  The 
plane  trigonometry  was  prepared  by  himself;  the  spherical  was  Eobert 
Simson's.  N"o  record  has  been  found  as  to  the  trigonometry  used  prior 
to  1822,  though  Simson's  was  probably  the  one.    It  does  not  appear  that 

*  See  also  an  article  by  Professor  Love  in  the  Nation  for  August  16,  1888. 


80  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

the  stndy  and  use  of  logaritlims  was  introduced  until  1811.  Natural 
philosophy  and  astronomy  were  taught  from  the  beginning.  Ferguson's 
text  was  the  one  first  used.  In  natural  philosophy  Nicholson's  was 
used  down  to  1809,  then  Helsham's  until  1816. 

Dr.  Caldwell  was  T)rofessor  of  mathematics  from  1796io  1817,  but  his 
activity  extended  in  many  other  directions.  He  "  taught  mathematics, 
natural  and  moral  philosophy,  and  did  all  the  preaching."  An  inter- 
esting, though  one-sided,  picture  of  him  as  a  teacher  of  geometry 
(about  the  year  1810)  is  given  by  William  Hooper,  one  of  the  alumni : 
"  There  being  but  three  teachers  in  college  (president,  professor  of  lan- 
guages, and  tutor),  the  Seniors  and  Juniors  had  but  one  recitation  per 
day.  The  Juniors  had  their  first  taste  of  geometry,  in  a  little  element- 
ary treatise,  drawn  up  by  Dr.  Caldwell,  in  manuscript,  and  not  then 
finished.  Copies  were  to  be  had  only  by  transcribing,  and  in  process 
of  time,  they,  of  course,  were  swarming  with  errors.  But  this  was  a 
decided  advantage  to  the  Junior,  who  stuck  to  his  text,  without  mind- 
ing his  diagram.  For,  if  he  happened  to  say  the  angle  at  A  was  equal 
to  the  angle  at  E,  when  in  fact  the  diagram  showed  no  angle  at  B  at 
all,  but  one  at  (7,  if  Dr.  Caldwell  corrected  him,  he  had  it  always  in  his 
power  to  say,  "  Well,  that  was  what  I  thought  myself,  but  it  ain't  so 
in  the  book^  and  I  thought  you  knew  better  than  I."  We  may  well 
suppose  that  the  doctor  was  completely  silenced  by  this  unexpected 
application  of  the  argu^nentum  ad  hominem.  *  *  *  The  Junior  hav- 
ing safely  got  through  with  his  mathematical  recitation  at  eleven  o'clock, 
was  free  till  the  next  day  at  the  same  hour."*  It  will  be  remembered, 
that  the  blackboard — that  simple  machine  which  doubles  the  teaching 
power  of  an  instructor  in  geometry — was  then  unknown  in  America. 

Fluxions  were  not  taught  at  that  time.  William  Hooper  says  in  bis 
humorous  way,  "As  for  chemistry  and  differential  and  integral  calculus, 
and  all  that,  we  never  heard  of  such  hard  things.  They  had  not  then 
crossed  the  Eoanoke,  nor  did  they  appear  among  us  till  they  were 
brought  in  by  the  northern  barbarians  about  the  year  1818."t  These 
northern  barbarians  were  Elisha  Mitchell  and  Denison  Olmsted.  The 
latter  introduced  chemistry,  mineralogy,  and  geology  into  the  univer- 
sity. Dr.  Mitchell  was  a  New  Euglander.  He  graduated  at  Yale  in  the 
class  of  1813  with  Olmsted.  He  began  teaching  immediately  after  grad- 
uation, and  in  1816  was  appointed  tutor  at  Yale.  At  the  University  of 
North  Carolina  he  held  the  chair  of  mathematics  from  1817  to  1825,  and 
performed  his  duties  with  energy  and  success.  When  Dr.  Olmsted  was 
called  to  Yale,  he  assumed  the  vacant  chair  of  chemistry,  which  posi- 
tion he  filled  with  great  credit  until  his  death  in  1857.  He  lost  his  life 
by  falling  over  a  precipice,  in  the  darkness,  while  engaged  in  the  scien- 
tific exploration  of  Mitchell's  Peak  in  western  North  Carolina. 

In  1818,  after  the  arrival  of  Mitchell,  spherical  trigonometry,  conic 
sections,  and  fluxions  were  introduced  into  the  course  of  study  leading 

'Fifty  Years  Since,  p.  23.  t  Page  17  of  his  address. 


INFLTTX   OP   ENGLISn   MATHEMATICS.  81 

to  the  degree  Df  A.  B.  The  course  was  as  follows :  Freshman  year,  arith- 
metic completed,  algebra  begun  j  Sophomore  year,  algebra  completed, 
geometry  5  Junior  year,  plane  trigonometry,  logarithms,  mensuration, 
navigation,  spherical  trigonometry,  conic  sections,  fluxions ;  Senior  year, 
astronomy,  natural  philosophy.  It  will  be  noticed  that  the  course  began 
now  in  the  Freshman  instead  of  the  Sophomore  year,  as  formerly.  If  it 
was  faithfully  carried  out,  then  it  must  have  been  very  creditable  to  the 
institution  at  that  time.  It  remained  nearly  unchanged  for  seventeen 
years.  As  regards  the  text- books,  it  is  probable  that  Simpson's  Al- 
gebra was  used  by  Mitchell ;  also  Button's,  and  since  1822,  Caldwell's 
Geometry  and  Trigonometry,  and  Vince's  Conic  Sections.  In  1823, 
Day's  work  on  mensuration  was  taught.  I^o  record  has  been  found  as 
to  what  text-books  were  used  when  fluxions  were  first  introduced.  It 
is  possible,  however,  that  Vince's  and  Hutton's  were  the  ones.  In  as- 
tronomy Nicholson's  was  used  for  a  long  time.  Cavallo's  I^atural  Phi- 
losophy and  Wood's  Mechanics  were  used,  the  latter  since  1822. 

Mr.  Paul  C.  Cameron  gives  an  interesting  reminiscence  of  B.  F. 
Moore,  a  once  prominent  lawyer.  "Often  has  he  entertained  me,"  says 
Cameron,  "  with  the  lights  and  shades  of  his  college  life;  how  grandly 
he  marched  through  the  recitations  in  the  languages  taught  in  the  first 
and  second  years  of  his  college  life ;  how  deep  and  suddenly  he  went 
under  when  he  struck  the  mathematical  course  of  the  Junior  year ;  how 
he  wrote  to  his  father  and  appealed  to  him  to  take  him  home  and  x)lace 
him  behind  the  plow.  His  father  refuses,  and  tells  him  to  make  known 
his  difficulties  to  his  professor.  He  hands  his  father's  letter  to  Dr. 
Mitchell,  who  invites  him  to  his  study  and  gives  him  instruction  by  the 
use  of  his  knife  and  a  piece  of  white  pine,  cutting  for  him  blocks  of 
mathematical  figures,  to  be  used  in  the  demonstrations  of  his  proposi- 
tions. Turning  the  light  on  him  in  this  way,  he  was  enabled  to  con- 
tinue his  course  with  satisfaction." 

UNIVERSITY  OF   SOUTH  CAROLmA.* 

The  South  Carolina  College  threw  open  its  doors  for  students  in  Jan- 
uary, 1805.  The  first  mathematical  teacher  at  the  college  was  Elisha 
Hammond  of  Massachusetts.  He  was  a  graduate  of  Dartmouth  Col- 
lege, and  when  called  to  this  position,  was  principal  of  Mt.  Bethel 
Academy  in  Kewberry,  S.  C.  After  one  year's  service  he  resigned  and 
returned  to  the  academy.  Judge  Evans,  a  student  under  him,  says 
"  His  personal  appearance  and  manners  were  very  captivating,  and  his 
popularity  for  the  period  of  his  connection  with  the  college  was  scarcely 
inferior  to  that  of  Dr.  Maxey."    Dr.  Maxey  was  the  president. 

Eev.  Joseph  Caldwell,  the  father  of  the  University  of  N"orth  Carolina, 
was  then  invited  to  the  chair  of  mathematics,  but  he  declined.  Paul 
H.  Perrault  was  elected  to  the  place,  but  in  1811  he  was  removed  for 

*  For  the  larger  part  of  our  information  respecting  this  institution,  we  are  indebted 
to  Professor  E.  W.  Davis,  professor  of  mathematics  and  astronomy  at  the  university. 
881— No.  3 6 


82  TEACHING   AND   HISTOEY   OF  MATHEMATICS. 

"neglect  of  college  duties."  He  is  said  to  have  been  "  well  skilled  in 
mathematics,"  but  "  wanting  in  that  dignity  which  a  Freshman  would 
expect  in  a  learned  professor."  After  his  separation  from  the  college 
he  was  attached  to  the  Army  as  a  topographical  engineer. 

The  mathematical  professor  for  the  next  four  years  was  George  Black- 
burn. He  was  a  graduate  of  Trinity  College,  Dublin.  He  taught  in  a 
military  academy  in  Philadelphia;  afterward  he  was  teacher  in  Virginia, 
and  was  then  called  to  the  chair  of  mathematics  and  astronomy  in  Will^ 
lam  and  Mary  College.  Thence  he  went  to  the  South  Carolina  College. 
In  1812  he  was  employed  by  the  State  of  South  Carolina  to  run  the 
boundary  line  between  North  and  South  Carolina.  An  old  student 
says  of  him :  "  He  was  a  man  of  quick  and  vigorous  understanding,  an 
able  mathematician,  and  a  most  excellent  instructor."  Another :  "  Pro- 
fessor Blackburn  was  a  first-rate  mathematician ;  he  taught  mathemat- 
ics as  a  science,  and  not  as  a  matter  of  memory.  From  him  I  learned 
the  demonstration  of  many  difficult  problems ;  and  with  his  aid  I  under- 
stood much  of  that  abstruse  and  difficult  science  as  applied  to  natural 
philosophy  and  astronomy."  He  made  students  think.  Whatdetracted 
somewhat  from  his  power  as  a  teacher  was  his  irritability. 

In  the  better  colleges  of  that  day,  the  curriculum  in  mathematics  em- 
braced a  short  course  on  fluxions,  or  calculus.  Though  the  plans  of 
study  included  then  about  all  the  subjects  pursued  in  the  average 
American  college  of  to-day,  these  subjects  were  not  taught  with  the 
same  thoroughness.  Moreover,  we  are  now  teaching  at  least  twice  as 
much  under  each  branch  as  was  taught  at  the  beginning  of  this  century. 
In  consequence  of  this,  students  of  former  times  began  the  study  of 
fluxions  when,  for  lack  of  prepatory  drill  in  lower  branches,  they  were 
far  less  able  to  wrestle  with  the  difficulties  of  the  transcendental  analysis 
than  are  our  students  of  to-day.  Professor  Blackburn's  teaching  of  the 
calculus,  as  narrated  by  M.  La  Borde,  in  his  History  of  South  Carolina 
College  (p.  82),  presents  a  picture  of  a  Senior  class  vainly  struggling 
with  the  intricacies  of  this  subject.  The  class  lost  interest  in  the  study 
and  was  very  remiss  in  its  attendance  upon  him,  and  those  who  did 
attend  failed  so  completely  in  unraveling  the  mysteries  of  the  transcen- 
dental analysis,  as  to  force  from  the  lips  of  the  professor  the  remark, 
"  that  it  might  be  that  half  of  his  class  were  very  smart  fellows,  for  he 
never  saw  themj  but  the  half  who  attended  his  recitations  were  as /la- 
borious as  oxen,  but  as  stupid  as  asses."  It  need  hardly  be  said  that 
this  was  the  cause  of  a  students'  rebellion. 

After  leaving  the  college,  Professor  Blackburn  made  latitude  and 
longitude  observations  for  the  State  map,  under  Governor  Allston. 
Later  he  settled  in  Baltimore,  where,  with  Dr.  Jennings,  he  founded 
Asbury  College.    His  last  days  were  spent  in  Columbia,  S.  C. 

From  1815  to  1820,  Christian  Hanckel,  a  Philadelphian  and  graduate 
of  the  University  of  Pennsylvania  (class  of  1810),  was  professor  of 
matb'^matics.    He  took  holy  orders  at  St.  Michael's,  Charleston.    His 


INFLUX    OF    ENGLISH   MA.THEMATICS.  83 

main  inducement  to  accept  the  chair  was  the  chance  to  build  up  the 
Protestant  Episcopal  Church  in  Columbia.  On  leaving  the  college,  he 
went  to  St.  Paul's  Church,  Charleston. 

The  requirements  for  admissiion  were,  according  to  catalogue,  at  the 
beginning,  "arithmetic,  including  proportion."  This,  most  probably, 
did  not  include  fractions.  In  1836  the  terms  were  "  arithmetic,  includ- 
ing fractions  and  the  extraction  of  roots. " 

In  the  earliest  course  of  mathematics  at  this  college,  the  Freshmen 
took  up  arithmetic ;  the  Sophomores,  common  and  decimal  fractions  with 
extraction  of  roots ;  the  Juniors,  geometry,  and  theoretical  and  practi- 
cal astronomy ;  the  Seniors,  exercises  in  higher  mathematics  as  directed 
by  the  faculty.  We  are  not  certain  that  this  curriculum  embraced  alge- 
bra. If  taught  then,  it  was  a  Senior  study.  Fractions  were,  it  seems, 
not  only  not  required  for  admission,  but  were  not  studied  before  the 
Sophomore  year. 

The  course  for  the  year  1811  was  considerably  stronger.  The  Fresh- 
men were  instructed  in  vulgar  and  decimal  fractions,  with  extraction 
of  roots ;  the  Sophomores  had  lectures  on  algebra ;  the  Juniors  studied 
Hutton's  course  of  mathematics ;  the  Seniors  had  lectures  by  the  "  pro- 
fessor of  mathematics,  mechanical  philosophy,  and  astronomy. "  From 
the  anecdote  told  of  Professor  Blackburn,  we  know  that  at  this  time,  or 
soon  after,  "  calculus "  (probably  fluxions)  was  taught  in  the  fourth 
year. 

KENTUCirr  UNIVERSiTY. 

About  the  year  1785  was  opened  in  Lincoln  County,  Kentucky,  a 
school  called  the  Transylvania  Seminary.  Four  years  later  it  was 
moved  to  Lexington,  Fayette  County,  where,  in  1790,  was  held  "  the  first 
public  college  commencement  in  the  West  of  which  we  have  any  record." 
On  January  1, 1799,  the  Transylvania  Seminary  and  a  similar  school, 
called  the  Kentucky  Academy,  were  united  under  the  name  of  Transyl- 
vania University.  The  Transylvania  University  existed  under  this 
name  until  1865,  when  it  was  merged  in  Kentucky  University,  and  the 
consolidation  has  since  been  conducted  under  the  name  and  charter  of 
the  latter. 

Little  has  been  done,  in  the  past,  to  preserve  the  history  of  these  in- 
stitutions. Some  of  the  records  appear  to  have  been  lost,  and  those  that 
are  stiil  extant  give  but  little  general  information.  The  data  on  the 
very  special  subject  of  mathematical  teaching  are  exceedingly  meagre. 
The  little  information  we  are  about  to  give  was  kindly  furnished  us  by 
President  Chas.  Louis  Loos,  of  Kentucky  University. 

The  records  of  Transylvania  University  Khow  that  on  September  16, 
1799,  "  mathematics  "  was  one  of  the  subjects  taught.  On  October  18, 
of  the  same  year,  the  following  books  are  mentioned  in  the  mathemati- 
cal course :  First  year,  "  Geography  by  Guthrie  or  Morse ;  algebra  by 
Saunderson,  Simson's  Euclid,  six  books;  trigonometry  and  mensura- 
tion, Gibson  j  N'avigatiou,  Patoun  or  Morse ;  Simson's  conic  sections." 


84  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

Second  year,  "Natural  pliilosophy  and  astronomy,  Fer^ison."  Tliese 
data  are  by  no  means  destitute  of  interest.  They  show  from  what 
sources  the  young  mathematician  "  in  the  West "  drew  his  intellectual 
food,  in  early  days.  On  October  26,  1799,  Eev.  James  Blythe  was 
elected  professor  of  "  science,"  which  term  was  made  to  include  mathe- 
matics. In  1803  the  professor  of  science  (J.  Blythe)  is  called  professor 
of  mathematics  and  natural  philosophy,  and  his  duties  were  to  teach 
"  geography,  arithmetic,  algebra,  geometry,  surveying,  navigation, 
conic  sections,  natural  philosophy,  and  astronomy."  In  1805  the  course 
was  the  same  as  the  one  just  given,  except  that  geography,  arithme- 
tic, and  surveying  are  not  mentioned. 

The  entry  in  the  records  for  March,  1S16,  gives  the  following  course 
in  mathematics : 

Freshmen,  first  six  books  of  Euclid,  plane  trigonometry,  surveying, 
navigation,  geography  j  Juniors,  algebra  as  far  as  affected  equations, 
spherical  trigonometry,  conic  sections,  natural  philosophy,  ancient 
geography  5  Seniors,  astronomy."  In  1817  Webber's  mathematics  is 
mentioned  as  a  text-book. 

THE  UNITED   STATES  MILITARY  ACADEMY. 

The  germinal  idea  of  the  United  States  Military  Academy  was  put 
forth  by  George  Washington,  who  felt,  probably  more  than  any  one 
else,  the  necessity  of  having  accomplished  engineers  in  time  of  war. 
The  Military  Academy  was  established  by  Congress  in  1802.  The  act 
was  limited  in  its  provisions  and  did  not  raise  the  academy  above  a 
military  post,  where  the  officers  of  engineers  might  give  or  receive  in- 
struction when  not  on  other  duty.  The  major  of  engineers  was  superin- 
tendent, the  two  captains  were  instructors,  and  the  cadets  were  pupils.* 

The  major  was  Jonathan  Williams  j  the  two  captains  were  William 
H.  Barron  and  Jared  Mansfield. 

Major  Williams,  in  a  report  to  the  Government  in  1808,  gives  us  some 
notion  of  the  early  instruction  at  the  academy.  He  says  that  "  The 
major  occasionally  read  lectures  on  fortifications,  gave  practical  lessons 
in  the  field,  and  taught  the  use  of  instruments  generally.  The  two 
caiJtaius  taught  mathematics,  the  one  in  the  line  of  geometrical,  the 
other  in  that  of  algebraical  demonstrations."  Mansfield  taught  also 
natural  philosophy.  He  had  previously  been  teacher  of  mathematics 
and  navigation  at  New  Haven,  and  then  at  Philadelphia.  He  had  pub- 
lished "essays"  of  some  originality  on  mathematics  and  physics. 
They  fell  under  the  notice  of  Thomas  Jefferson,  and  were  the  means 
that  led  to  his  appointment  by  the  President  as  captain  of  engineers 
for  the  very  purpose  of  becoming  teacher  at  West  Point.  But  after  one 
year's  teaching  he  was  appointed  by  Jefierson^  in  1805,  to  establish 
meridian  lines  and  base  lines  in  the  North- West  Territory  for  the  pur- 

•The  U.  S.  Military  Academy  at  West  Point,  by  Edward  D.  Mansfield,  LL.  D. 


INFLUX    OP   ENGLISH    MATHEMATICS.  85 

pose  of  public  surveySo    His  position  remained  vacant  until  his  return, 
after  the  War  of  1812. 

In  1806  Alden  Partridge  became  assistant  in  mathematics.  He  was 
a  native  of  Vermont,  had  entered  Dartmouth  College  in  1803,  but  be- 
fore completing  his  course  became  cadet  at  West  Point. 

Captain  Barron  was  relieved  in  1807  by  Ferdinand  E.  Hassler,  who 
continued  there  until  1810,  when  he  resigned.  The  following  year  he 
was  called  to  the  United  States  Coast  Survey.  Hassler  was  a  Swiss. 
It  was  again  the  keen  eye  of  President  Jefferson  that  recognized  the 
talent  and  secured  the  services  of  this  foreigner,  who  had  shortly  before 
landed  on  our  shores.  Hassler's  teaching  power  must  have  been  ham- 
pered somewhat  by  his  limited  acquaintance  with  the  English  lan- 
guage. While  at  West  Point  he  began  writing  his  "Elements  of  An- 
alytic Trigonometry,"  published  by  him  in  1826.  It  was  written  in 
French  and  then  translated  for  publication  by  Professor  Eenwick. 
From  its  preface  we  take  the  following:  "It  was  the  desire  of  intro- 
ducing into  the  course  of  mathematics  at  West  Point  the  most  useful 
mode  of  instruction  in  this  branch  that  led  me  to  the  preparation  of 
this  work  as  early  as  the  year  1807."  Hassler  was,  no  doubt,  the  first 
one  to  teach  analytic  trigonometry  in  this  country — the  first  one  to  dis- 
card the  old  '<^  line-system." 

About  the  same  time  Christian  Zoeller,  also  a  Swiss,  was  made  in- 
structor in  drawing.  He  was  "an  amiable  man  of  no  high  attain- 
ments." 

Down  to  the  year  1812  the  academy  was  in  a  chaotic  condition. 
There  was  no  regular  corps  of  instwictors,  and  no  regular  classes. 
There  had  been  no  continuous  study  of  any  subject  except  mathe- 
matics. Eeferring  to  Hassler,  Major  Williams  says  in  his  report  of  1808, 
"  During  the  last  year  a  citizen  of  eminent  talents  as  a  mathematician 
has  been  employed  as  principal  teacher,"  and  "  being  the  only  teacher 
designated  by  the  law,  he  is  the  only  one  that,  exclusive  of  the  corps 
of  engineers,  can  be  said  to  belong  to  the  institution,"  In  conclusion 
the  major  says :  "In  short,  the  Military  Academy  as  it  now  stands  is 
like  a  foundling,  barely  existing  among  the  mountains,  nurtured  at  a 
distance  out  of  sight,  and  almost  unknown  to  its  legitimate  parents." 

In  vain  did  Jefferson  in  1808  and  Madison  in  1810  recommend  to 
Congress  the  enlargement  of  the  academy.  It  was  not  until  the  nation 
was  roused  by  the  shock  of  war  that  Congress  began  to  act.  In  1812 
Congress  made  liberal  appropriations  and  passed  an  act  reorganizing 
the  institution.  The  next  five  years  are  the  formation  period  of  the 
academy.  The  first  reform  to  be  accomi3lished  was  the  placing  of  the  in- 
struction on  a  higher  level.  The  first  academic  faculty  was  constituted 
as  follows  :  Col.  Jared  Mansfield,  profess()r  of  natural  and  experimental 
philosophy ;  Andrew  Ellicott,  professor  of  mathematics ;  Alden  Part' 
ridge,  professor  of  engineering ;  Christian  Zoeller,  professor  of  drawing. 
We  see  from  this  that  Mansfield  held  now  the  same  place  as  in  1804, 
and  Partridge  was  promoted  from  assistant  to  the  rank  o£  professor. 


86  TEACHING   AFD  HISTORY    OF   MATHEMATICS. 

Mansfield  and  Bllicott  bad  long  been  in  tlie  service  of  the  General  Gov- 
ernment and  of  State  governments  in  the  capacity  of  surveyors  a^d 
astronomers,  and  had  established  a  wide  reputation  for  both  their  prac- 
tical and  theoretical  knowledge  of  mathematics.  But  now  they  were 
old  men,  and  their  ideas  were  somewhat  old-fashioned.  The  workings 
of  this  faculty  were  not  altogether  harmonious.  Partridge,  being 
strong-willed  and  eccentric,  wanted  to  have  everything  his  own  way. 
He  was  removed  from  his  place.  The  appointment  of  Major  Sylvanus 
Thayer,  in  1817,  to  the  superintendency  of  the  academy  marks  a  new 
era  in  its  history. 

Some  notion  of  the  instruction  in  mathematics  at  West  Point  between 
1812  and  1817  may  be  obtained  from  the  following  extract  from  the  cur- 
riculum which,  in  1816,  received  the  official  approval  of  the  Secretary  of 
War. 

'^MatliemaUcs. — A  complete  course  of  mathematics  shall  embrace  the 
following  branches,  namely :  The  nature  and  construction  of  logarithms 
and  the  use  of  the  tables ;  algebra,  to  include  the  solution  of  cubic 
equations,  with  all  the  preceding  rules ;  geometry,  to  include  plane  and 
solid  geometry,  also  ratios  and  proportions,  and  the  construction  of 
geometrical  problems,  ajjplication  of  algebra  to  geometry,  practical  geom- 
etry on  the  ground,  mensuration  of  planes  and  solids ;  plane  trigonom- 
etry, with  its  application  to  surveying  and  the  mensuration  of  heights 
and  distances  j  spherical  trigonometry,  with  its  application  to  the  solu- 
tion of  spherical  problems ;  the  doctrine  of  infinite  series^  conic  sections, 
with  their  application  to  projectiles;  fluxions,  to  be  taught  at  the  op- 
tion of  the  professor  and  student."^ 

There  was,  however,  no  instruction  in  fluxions.  E.  D.  Mansfield,  in 
his  historical  sketch  of  the  academy,  does  not  include  fluxions  in  the  cur- 
riculum for  181G,  but  he  remarks  that  calculus  was  added  to  the  course  a 
year  or  two  later.    The  text-book  then  in  use  was  Hutton's  Mathematics. 

Thus  far  the  cadets  were  admitted  to  the  academy  without  entrance 
examinations,  and  poor  results  were  reached.  Many  cadets  were  unfit 
by  prior  study  for  the  subjects  they  had  to  pursue.  Eank  and  assign- 
ment to  the  various  army  corps  were  not  made  to  depend  upon  merit.* 

Selp-taught  Mathematicians. 

The  foremost  American  mathematician  of  this  time,  like  David  Kit- 
tenhouse  and  Thomas  Godfrey,  had  not  enjoyed  the  privileges  of  a  col- 
lege education  ;  like  them,  he  was  self-taught.  We  have  reference  to 
Nathaniel  Bowditch.t 

*  The  College  Book,  edited  by  Charles  F.  Kichardsou  aud  Henry  A.  Clark,  p.  216. 

IThis  sketch  is  extracted  from  the  Memoirs  of  Nathaniel  Bowditch,  by  his  son, 
Nathaniel  I.  Bowditch  (Boston,  1839);  from  the  Discourse  on  the  Life  and  Character  of 
Nathaniel  Bowditch,  by  Alexander  Young  (Boston,  1833) ;  from  the  eulogy  by  Pro- 
fessor Pickering  (Boston,  1838),  and  from  the  eulogy  by  Judge  Daniel  A.  White 
(Salem,  1838).  A  full  list  of  Bowditch's  mathematical  papers  may  be  found  in  the 
Mathematical  Monthly,  Vol.  II. 


INFLUX   OF   ENGLISH  MATHEMATICS.  87 

It  is  instructive  to  study  tlie  history  of  Ms  early  life  and  to  ascertain 
the  influences  under  which  his  mind  was  formed.  He  was  born  at 
Salem,  Mass.,  in  1773.  His  parents  were  poor,  and  he  had  often  to  con- 
tent himself  with  a  dinner  consisting  chiefly  of  potatoes,  and  at  near 
approach  of  winter  to  continue  wearing  the  thin  garments  of  summer. 
After  attending  for  a  short  time  a  dame's  school  near  Salem,  he  en- 
tered Wajtson's  school,  which  was  the  best  school  in  Salem.  It  was 
wholly  inadequate  to  furnish  the  ground-work  and  elements  of  a  re- 
spectable education.  He  entered  the  school  at  the  age  of  seven  and 
remained  there  three  years. 

Bowditch  early  showed  a  great  fondness  for  mathematics ;  but  on 
account  of  his  extreme  youth  his  master  refused  to  admit  him  to  this 
study  until  he  had  procured  from  his  father  a  special  request  to  that 
effect.  On  one  occasion  he  solved  a  problem  in  arithmetic  which  the 
instructor  thought  must  be  far  above  his  comprehension.  On  being 
asked  who  had  been  doing  the  sum  for  him  he  answered,  "Kobody — I 
did  it  myself."  He  was  then  accused  of  falsehood  and  treated  with 
much  severity. 

When  he  was  ten  years  old  he  left  school  to  work  in  the  shop  of  his 
father,  who  was  a  cooper.  He  received  no  regular  instruction  after 
leaving  school,  excepting  a  few  lessons  in  book-keeping.  He  became 
soon  after  an  apprentice  to  a  ship-chandler,  and  afterward  was  clerk 
in  a  large  mercantile  establishment.  It  was  during  his  apprenticeship 
that  he  disclosed  that  strong  bent  for  mathematical  studies.  Every 
moment  that  he  could  snatch  from  the  counter  was  given  to  the  slate. 
When  he  was  only  fifteen  years  old  he  made  an  almanac  for  the  year 
1790,  containing  all  the  usual  tables,  calculations  of  the  eclipses  and 
other  phenomena,  and  even  the  customary  predictions  of  the  weather. 

When  he  was  fourteen  years  old  he  one  day  got  from  an  elder  brother 
a  vague  account  of  a  method  of  working  out  problems  by  letters  instead 
of  figures.  This  novelty  excited  his  curiosity ;  he  succeeded  in  bor- 
rowing an  algebra,  and  "  that  night,"  says  he,  "  I  did  not  close  my 
eyes."  He  read  it,  and  read  it  again,  and  mastered  its  contents  j  and 
copied  it  out  from  beginning  to  end. 

Subsequently  he  acquired  access  to  an  extensive  scientific  library  of 
Dr.  Eichard  Kirwan,  an  Irish  scientist,  which  had  been  captured  in  the 
British  channel  by  a  privateer  and  sold  to  a  society  of  gentlemen  at 
Salem.  This  became  the  basis  of  the  present  Salem  Atheneum.  He 
found  there  the  Philosophical  Transactions  of  the  Eoyal  Society  of  Lon- 
don, from  which  he  made  full  and  minute  abstracts  of  the  mathematical 
papers  contained  in  them.  At  this  time  he  was  too  poor  to  buy  books, 
and  this  was  the  only  way  in  which  he  could  manage  to  have  them  for  con- 
venient reference.  The  title  page  of  one  of  these  manuscript  volumes 
states  that  it  contains,  with  the  next  volume,  "  A  complete  collection  of 
all  the  mathematical  papers  in  the  Philosophical  Transactions ;  extracts 
from  various  encyclopedias  j  from  the  Memoirs  of  the  Paris  Academy,- 


88  TEACHING   AND   HISTORY    OP   MATHEMATICS. 

a  complete  copy  of  Emerson's  Mechanics  5  a  copy  of  Hamilton's  Conies; 
extracts  from  Graresande's  and  Martin's  Philosophical  Treatises ;  from 
Bernoulli,  etc.,  etc."  What  perseverance,  what  energy,  what  enthusi- 
asm is  displayed  in  this  laborious  work  of  copying! 

Eowditch  was  very  fond  of  books,  but  having  no  guide  in  the  selection 
of  them  his  reading  was  at  first  of  the  most  miscellaneous  character. 
Thus  he  read  every  article  in  Chambers'  Encyclopaedia  from  beginning  to 
end.  He  secured  a  copy  of  Newton's  Principia,  but  as  it  was  published 
in  Latin  he  began  the  study  of  that  language  that  he  might  read  that 
great  work.  By  great  perseverance  he  learned  enough  Latin  to  enable 
him  to  read  any  work  of  science  in  it.  He  afterward  learned  French 
for  the  purpose  of  having  access  to  the  treasures  of  French  mathematical 
science,  and  at  a  late  period  of  his  life  he  acquired  some  knowledge  of  the 
German  language.  When  twenty-one  years  of  age  he  had  read  the  im- 
mortal work  of  Kewton,  and  there  were  few  in  his  State  who  surpassed 
him  in  mathematical  attainments. 

Eowditch  did  not  long  remain  in  the  situation  of  a  merchant's  clerk. 
His  mathematical  talent,  in  a  town  distinguished  for  enterprise,  could 
not  fail  of  being  called  into  exercise  in  connection  with  the  art  of  navi- 
gation. He  became  a  practical  navigator.  Between  1795  and  1804  he 
was  on  five  sea  voyages,  ail  under  the  command  of  Captain  Henry  Prince, 
of  Salem. 

The  leisure  of  the  long  Bast  India  voyages,  when  the  ship  was  lazily 
sweeping  along  under  the  steady  impulse  of  the  trade  winds,  afforded 
him  fine  opportunities  for  pursuing  his  mathematical  studies,  as  well  as 
for  indulging  his  taste  in  general  literature.  The  French  mathemati- 
cian Lacroix  acknowledged  to  a  young  American  that  he  was  indebted 
to  Mr.  Bowditch  for  communicating  many  errors  in  his  works,  which  he 
had  discovered  in  these  same  long  India  voyages.  It  was  his  practice 
both  when  at  home  and  when  at  sea  to  rise  at  a  very  early  hour  in  the 
morning  and  pursue  his  studies.  He  was  ofteii  seen  on  deck  "walking 
rapidly  and  apparently  in  deep  thought,  when  it  was  well  understood 
by  all  on  board  that  he  was  not  to  be  disturbed,  as  we  supposed  he  was 
solving  some  diflicult  problem,  and  when  he  darted  below  the  conclu- 
sion was  that  he  had  got  the  idea  5  if  he  were  in  the  fore  part  of  the  ship 
when  the  idea  came  to  him,  he  would  actually  run  to  the  cabin,  and  his 
countenance  would  give  the  expression  that  he  had  found  a  prize." 

"He  loved  to  study  himself,"  says  Captain  Prince,  "  and  he  loved  to 
see  others  study.  He  was  always  fond  of  teaching  others.  He  would 
do  anything  if  any  one  would  show  a  disposition  to  learn.  Hence,"  he 
adds,  "  all  was  harmony  on  board;  all  had  a  zeal  for  study ;  all  were 
ambitious  to  learn."  On  one  occasion  two  sailors  were  zealously  dis- 
puting, in  the  hearing  of  the  captain  and  supercargo,  respecting  sines 
ahd  co-sines.  As  a  result  of  his  teaching,  the  whole  crew,  yea,  even  the 
negro  cook,  acquired  the  knowledge  of  how  to  compute  a  lunar  obser- 
vation.   When  the  capta|n  once  arrived  at  Manila,  he  was  asked  iiow  he 


INFLUX    OF   ENGLISH   MATHEMATICS.  89 

contrived  to  find  his  way,  in  the  face  of  a  northeast  monsoon,  by  mere 
dead  reckoning.  He  replied,  that  "  he  had  a  crew  of  twelve  men  every 
one  of  whom  could  take  and  work  a  lunar  observation  as  well  for  all 
practical  j)urposes  as  Sir  Isaac  Kewton  himself,  were  he  alive."  Dur- 
ing this  conversation  Bowditch  sat  "  as  modest  as  a  maid,  saying  not 
a  word,  but  holding  his  slate  pencil  in  his  mouth ; "  while  another 
person  remarked,  that  "  there  was  more  knowledge  of  navigation  on 
board  that  ship  than  there  ever  was  in  all  the  vessels  that  have  floated 
in  Manila  Bay. 

At  that  period  the  common  treatise  on  navigation  was  the  well  known 
work  of  Hamilton  Moore,  which  had  occasioned  many  a  shipwreck,  but 
which  Bowditch,  like  other  navigators,  was  obliged  to  use.  He  found 
it  abounding  with  blunders  and  overrun  with  typographical  errors. 
Of  these  last  errors  many  thousands  of  more  or  less  importance  were 
corrected  in  the  early  revisions  of  the  work.  Bowditch  published  sev. 
eral  editions  of  Moore's  works  under  that  author's  name,  but  the  whole 
book  at  length  underwent  so  many  changes  and  radical  improvements 
as  to  justify  him  to  take  it  out  on  his  own  name.  This  is  the  origin 
of  Bowditch's  Practical  ii^avigator,  the  best  book  on  navigation  then 
in  existence.  The  following  particulars  regarding  the  publication  of 
this  work  have  been  handed  down  to  us  : 

t  The  first  American  edition  was  printed  in  1801,  but  not  published  until 
1802.  The  publisher,  Mr.  Blunt,  took  the  work  and  a  copy  of  Hamilton 
Moore,  with  all  the  errors  marked,  to  England,  called  on  the  publishers 
of  Hamilton  Moore,  and  sold  the  printed  copy  of  Bowditch  on  condi- 
tion that  the  American  edition  should  not  be  sold  until  June,  1802,  to 
give  them  an  opportunity  to  get  theirs  into  the  English  market  at  the 
same  time.  The  London  edition  was  revised  and  newly  arranged  by 
Thomas  Kirby,  teacher  of  mathematics  and  nautical  astronomy.  It 
was  recommended  as  an  improvement  on  Bowditch,  but  it  contained 
many  errors.  This  gave  occasion  to  a  British  writer,  Andrew  Mackay, 
who  published  a  rival  work  on  navigation,  to  make  Dr.  Bowditch's  sup- 
posed inaccuracies  a  particular  object  of  attack.*  This  charge  was  em- 
phatically repelled  by  Bowditch  in  the  new  edition  of  1807. 
;  From  Harvard  College  Bowditch  received  the  highest  encouragement 
to  pursue  his  scientific  studies.  In  July,  1802,  when  his  ship  was  wind- 
bound  in  Boston,  he  went  to  attend  the  commencement  exercises  at 
Harvard  j  and  among  the  honorary  degrees  conferred,  he  thought  he 
heard  his  own  name  announced  as  a  master  of  arts;  but  it  was  not  till 
congratulated  by  a  friend  that  he  became  satisfied  that  his  senses  had 
not  deceived  him.  He  always  spoke  of  this  as  one  of  the  proudest 
days  of  his  life,  and  amid  all  subsequent  distinctions  conferred  upon 
him  from  foreign  countries,  he  recurred  to  this  with  greatest  pleasure. 

*  Memoirs  of  American  Academy  of  Arts  aad  Science,  Vol.  II,  1846,  Eulogy  on 
Bowditch,  note  C. 


90  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

On  quittiog  the  sea,  in  1803,  he  was  appointed  president  of  an  in- 
surance company  in  Salem,  the  duties  of  which  he  continued  to  dis- 
charge for  twenty  years,  when  he  accepted  the  position  of  actuary  of 
the  Massachusetts  Hospital  Life  Insurance  (3ompauy  in  Boston.  For 
many  years  he  discharged  the  duties  of  this  of&ce  with  the  greatest 
fidelity  and  skill. 

He  was  several  times  solicited  to  accept  positions  in  various  literary 
institutions.  In  1806  he  was  chosen  to  fill  the  Hollis  Professorship  of 
Mathematics  at  Harvard.  He  received  from  Thomas  Jefferson  the  offer 
of  the  professorship  of  mathematics  at  the  University  of  Virginia. 
Jefferson  said  in  his  letter :  "  We  are  satisfied  we  can  get  from  no  coun- 
try a  professor  of  higher  qualifications  than  yourself  for  our  mathe- 
matical department."  In  1820  he  was  asked  to  permit  his  name  to  be 
presented  to  the  President  of  the  United  States  to  fill  a  vacant  chair 
at  the  U.  S.  Military  Academy  at  West  Point.  Bowditch  could  not  be 
persuaded  to  accept  any  of  these  positions. 

The  work  for  which  Bowditch  was  for  a  long  time  exclusively  known 
was  his  Practical  E"avigator.  This  gave  him  a  wide-spread  popularity 
among  sea  faring  people  everywhere.  Bowditch  himself  did  not  con- 
sider this  work  as  one  which  would  advance  his  scientific  reputation. 
What  established  his  celebrity  as  a  man  of  science  was  not  his  Practical 
Navigator,  but  his  translation,  with  a  commentary,  of  the  epoch-mak- 
ing work  of  Laplace,  called  the  Mecanique  Celeste. 

Later  on  we  shall  speak  of  this  translation  at  length.  Bowditch  con- 
tributed many  articles  to  the  American  edition  of  Eees's  Oyclopgedia. 

The  question  may  be  asked,  how  should  Bowditch  be  ranked  as  a 
mathematician  ?  In  answer  to  this  we  may  say,  that  he  is  acknowledged 
by  all  as  having  stood  at  the  head  of  scientific  men  of  this  country,  and 
to  have  contributed  more  to  his  country's  reputation  than  any  contem- 
porary scientist.  But  a  giant  in  Liliput  is  not  necessarily  a  giant  in 
another  country.  Though  a  man  of  great  energy  and  intellectual  powers, 
he  can  not  be  pronounced  a  first-class  mathematician.  He  was  a  man 
of  learning,  but  not  a  man  of  genius  in  the  sense  that  Newton,  Leibnitz, 
Gauss,  Abel,  Pascal,  and  Archimedes  were  men  of  genius.  The  esti- 
mate that  Bowditch  made  of  his  own  capacities  and  gifts  was,  in  our 
opinion,  accurate,  fair,  and  just.  He  did  not  overrate  his  talents,  nor 
did  he,  with  assumed  humility,  purposely  underrate  his  powers.  He  is 
reported  as  having  once  said,  "People  are  very  kind  and  polite,  in  men- 
tioning me  in  the  same  breath  with  Laplace,  and  blending  my  name 
with  his.  But  they  mistake  both  me  and  him  j  we  are  very  different 
men.  I  trust  I  understand  his  works,  and  can  supply  his  deficiencies, 
and  record  the  successive  advances  of  the  science,  and  perhaps  append 
some  improvements.  But  Laplace  was  a  genius,  a  discoverer,  an  in- 
ventor ;  and  yet  I  hope  I  know  as  much  about  mathematics  as  Playfair ! " 

The  career  of  Bowditch  furnishes  us  with  an  excellent  illustration  of 
how  much  may  be  accomplished  through  indefatigable  energy  and  per- 


INFLUX    OF   ENGLISH   MATHEMATICS.  91 

servance  by  a  mind  which,  though  naturally  far  above  the  average 
mind,  is,  nevertheless,  lacking  the  powers  of  real  genius. 

A  mathematician  of  considerable  local  reputation  was  Enoch  Lewis 
(1776-1856).  He  was  a  native  of  Pennsylvania  and  a  Quaker.  In  1799 
he  became  teacher  of  mathematics  at  the  West  Town  Boarding  School, 
established  by  the  Society  of  Friends.  He  was  the  author  of  treatises 
on  arithmetic,  algebra,  and  trigonometry. 

Under  the  tuition  of  Enoch  Lewis,  for  six  months,  at  the  Friends' 
Boarding  School  at  West  Town,  was  John  Gummere,  who  was  then 
about  twenty  years  old.  Excepting  in  reading,  writing,  and  arithmetic, 
he  had  received  no  instruction  whatever  up  to  that  time.  After  teach- 
ing elementary  schools  for  six  years,  he  determined  in  1814  to  open  a 
boarding  school  in  Burlington.  The  following  story  characterizes  the 
young  man.* 

He  determined  to  give  courses  of  lectures  in  natural  philosophy  and 
chemistry,  and  proposed  .to  his  brother,  who  had  joined  him  in  the 
school,  that  he  should  take  the  latter.  The  brother  replied  that  he 
had  never  opened  a  book  on  chemistry.  "  K^either  have  I,"  said  John, 
"  on  natural  philosophy."  It  was  then  objected  that  they  could  not 
obtain  the  appropriate  instruments  and  apparatus  in  this  country. 
"  But  we  can  get  them,"  he  said,  "from  London."  It  was  suggested 
that  they  might  fail  in  so  making  themselves  masters  of  their  respect- 
ive subjects  as  to  pursue  them  advantageously.  '■'■  But  we  shall  not 
fail,"  said  he;  "only  determine  and  the  thing  is  half  done."  An  order 
was  sent  to  London  for  apparatus,  both  philosophical  and  chemical,  a 
better  supply  of  which  was  provided  for  his  institution  (at  an  expense 
of  several  thousand  dollars)  than  was  to  be  found  in  any  private  insti- 
tution in  this  country. 

Gummere  acquired  considerable  reputation  as  a  teacher  and  writer. 
He  was  for  over  forty  years  teacher  in  Friends'  schools  in  Pennsylvania 
and  New  Jersey.  He  once  declined  the  proffered  chair  of  mathematics 
at  the  University  of  Pennsylvania.  He  contributed  astronomical  papers 
to  the  American  Philosophical  Society.  The  most  celebrated  of  his 
works  was  his  Surveying  (1814),  which  went  through  a  large  number  of 
editions.  It  was  more  extensively  known  and  more  highly  prized  than 
any  other  work  on  surveying.  His  treatise  on  theoretical  and  practical 
astronomy  was  once  used  as  a  text-book  at  West  Point  and  other 
leading  scientific  institutions.  In  preparing  it  he  bad  greatly  profited 
by  French  models. 

Mention  should  be  made  here  of  the  mathematical  studies  of  Walter 
Folger,  a  lawyer,  of  Nantucket.  They  will  throw  light  upon  the  kind 
of  instruction  which  was  then  being  given  at  our  American  ports,  in 
iught  schools  for  navigators.  After  attending  common  schools,  Folger 
studied  land-surveying,  in  which,  without  the  least  assistance,  he  be- 
came exceedingly  skillful.     "  In  the  winter  of  1782-83  he  attended  an 

*  Memorials  of  the  Life  and  Character  of  John  Gummere,  by  William  J.  Allison. 


92  TEACHING    AND    HISTORY    OF    MATHEMATICS. 

evening  school  in  which  he  studied  navigation,  and  readily  acquainted 
himself  with  these  branches.  Nothing  of  a  mathematical  character 
seemed  ever  to  present  any  difficulties  to  his  mind.  He  mastered  al- 
gebra and  fluxions  without  assistance,  and  while  in  his  teens  he  read 
Euclid  as  he  would  read  a  narrative,  no  problem  arresting  his  progress ; 
and  yet,  so  little  did  he  know  of  language,  or  of  anything  appertaining 
to  it,  that  he  had  reached  the  years  of  manhood,  as  he  often  confessed, 
before  he  knew  the  definition  of  the  word  grammar.''^  * 

''His  father  finally  succeeded  in  obtaining  for  him  a  work  of  naviga- 
tion, to  which,  for  the  first  time,  was  appended  Dr.  Maskelyne's  method 
of  obtaining  the  longitude  at  sea  by  means  of  lunar  distances.  This 
deligbted  him,  and  at  the  age  of  eighteen,  [when]  prostrated  with  sick- 
ness, he  familiarized  himself  with  the  problem,  and  the  engagement  so 
diverted  his  mind  from  his  infirmities  that  he  speedily  regained  his 
strength.  He  immediately  applied  all  his  influence  to  the  encourage- 
ment of  the  use  of  this  method  among  his  fellow-townsmen,  then  univer- 
sally engaged  in  the  prosecution  of  whaling  voyages.  To  numbers  he 
gave  personal  instruction,  and  the  first  American  ship-master  who  de- 
termined his  longitude  by  lunar  observations  is  said  to  have  been  one 
of  his  pupils."  A  similar  school  was  held  in  Philadelphia  by  Eobert 
Patterson. 

Surveying  of  Government  Lands. 

Tn  a  new  and  growing  country  like  ours  it  was  only  natural  that  the 
art  of  surveying  should  have  been  early  cultivated.  But  to  a  surveyor 
some  knowledge  of  the  rudiments  of  geometry  and  trigonometry  was 
indispensable.  As  early  as  1761  there  was  published,  or  reprinted,  in 
Philadelphia  a  work  entitled,  Subtential  Plane  Trigonometry,  by  Thomas 
Abel,  presumably  an  English  teacher.  In  1785  there  was  reprinted  in 
Philadelpia  an  edition  of  Eobert  Gibson's  Practical  Surveying,  which 
first  appeared  in  London  in  1767.  This  enjoyed  an  extended  circulation. 
In  1799  appeared  in  Wilmington  the  first  popular  American  treatise  on 
surveying,  by  Zachariah  Jess,  a  teacher  and  practical  surveyor,  of  Dela- 
ware. In  the  pieface  to  Gummere's  Treatise  on  Surveying  (1814)  we 
read :  "  The  works  of  Gibson  and  Jess  are  the  only  ones  at  present  in 
general  use.  The  former,  though  much  the  better  of  the  two,  is  de- 
ficient in  many  respects."  In  1796  was  published  in  I^ew  York,  The  Art 
of  Surveying  Made  Easy,  by  John  Love,  and  at  Litchfield,  An  Accurate 
System  of  Surveying,  by  Samuel  Moore.  In  1806  Eev.  Abel  Flint  pub- 
lished his  Geometry  and  Trigonometry,  with  a  Treatise  on  Surveying. 
Flint  graduated  at  Yale  in  1785,  was  tutor  at  Brown  till  1790,  after- 
ward studied  theology,  and  then  became  pastor  at  Hartford,  Conn. 

The  publication  of  Gibson's  Surveying  in  1785  was  very  timely,  for 
it  was  in  this  very  year  that  Congress  passed  an  ordinance  specifying 

*  "A  Brief  Memoir  of  tlie  late  Walter  Folger,  of  Nantucket,"  by  William  Mitchell, 
in  tlie  American  Journal  of  Science  and  Arts,  second  series,  Vol.  IX,  No,  27,  May,  1850. 


INFLUX    OF    ENGLISH   MATHEMATICS.  93 

that  surveyors,  as  they  were  respectively  qualified,  should  proceed  to 
divide  the  western  territory  into  townships  of  6  miles  square  by  lines  run- 
ning due  north  and  south,  and  others  crossing  these  at  right  angles  as 
near  as  may  be.  Each  township  should  be  subdivided  into  lots  of  one 
mile  square.  This  system  was  not  universally  approved,  for  it  tended 
to  delay  the  sale  of  public  lands  till  they  could  be  correctly  measured. 
In  the  Madison  Papers  (Vol  II,  p.  C40)  we  read  that  the  Eastern  States 
favored  the  plan  adopted,  while  the  Southern  were  "  biased  in  favor  of 
indiscriminate  location."  Kentucky  and  Tennessee  adhered  to  the  old 
plan  of  indiscriminate  location.  This  occasioned  so  much  litigation  in 
those  States  that  it  has  been  said  that  as  much  money  was  annually 
expended  there  in  land-title  litigation  as  would  defray  their  taxes  for 
the  support  of  the  severest  war.  Lands  surveyed  by  the  United  States, 
on  the  other  hand,  were  comparatively  without  any  legal  difficulty.  In 
fact,  one  great  object  of  the  Government  system  was  the  removal  of  all 
temptation  to  incur  the  curse  pronounced  by  Moses  on  him  "  who  re- 
moveth  his  neighbor's  landmark."  The  corners  of  each  section  were 
carefully  located  by  marked  trees,  whose  species,  diameter,  distance, 
and  bearing  were  entered  upon  the  tield-notes.  If  the  marked  tree  at 
any  one  corner  were  destroyed,  then  its  location  could  be  determined 
from  the  other  corners.  Though  a  great  improvement  on  previous 
modes  of  surveying,  it  is  inaccurate  and  rude  indeed  as  compared  with 
the  refined  triangulation  surveys  now  carried  on  by  the  United  States 
Coast  and  Geodetic  Survey. 

Most  conspicuous  in  the  execution  of  the  early  Government  surveys 
were  Andrew  Ellicott  and  Jared  Mansfield.  EUicott  was  engaged  in 
a  large  number  of  surveys.  At  various  times  he  was  appointed  com- 
missioner for  marking  the  boundaries  of  Virginia,  Pennsylvania,  and 
I^ew  York ;  in  1789  he  was  selected  by  Washington  to  survey  the  land 
lying  between  Pennsylvania  and  Lake  Erie  j  in  1790  he  was  employed, 
with  his  brother  Joseph,  in  surveying  and  laying  out  the  city  of  Wash- 
ington; in  1792  he  was  made  Surveyor-General  of  the  United  States;  in 
1796  he  was  appointed  United  States  Commissioner,  under  the  treaty  of 
San  Lorenzo  el  Eeal,  to  determine  the  boundary  between  the  United 
States  and  the  Spanish  possessions  on  the  south.  It  is  stated  that  he 
sent  observations  to  Delambre,  of  France,  remarking  that  they  were 
made  by  a  "  self-taught  astronomer,  and  the  only  practical  one  now  in 
the  United  States."  This  was  after  the  death  of  David  Eittenhouse. 
.  More  prominently  connected  with  the  survey  of  the  Korth-West  Terri- 
tory than  Ellicott  was  Jared  Mansfield.  He  was  a  graduate  of  Yale 
College.  In  1801  (!)  he  published  Essays,  Mathematical  and  Physical. 
From  the  perusal  of  his  works  alone  the  illustrious  Thomas  Jefferson 
was  induced  to  bring  him  into  public  life.  In  1803  he  was  appointed 
surveyor-general  of  the  Korth-West  Territory.  His  first  work  was  to 
determine  astronomically  certain  lines  of  latitude  and  the  principal 
meridians  on  which\the  surveys  were  to  proceed.    To  carry  out  this 


94  TEACHING   AND   HISTORY   OF  MATHEMATICS. 

work  astronomical  instruments  were  needed.  President  Jefferson  or- 
dered the  purchase  from  London  of  a  transit  instrument,  a  telescope, 
an  astronomical  clock,  and  a  sextant.  The  first  principal  meridian  be- 
gan at  the  mouth  of  the  Great  Miami;  the  second  at  a  point  5  miles 
south-west  of  the  confluence  of  Little  Bliie  Eiver  with  the  Ohio ;  the 
third  at  the  confluence  of  the  Ohio  and  the  Mississippi  Elvers ;  the 
fourth  at  the  junction  of  the  Illinois  and  Mississippi;  the  fifth  at  the 
mouth  of  the  Arkansas  Eiver.  A  large  number  of  other  meridians,  or 
"  base-lines,"  have  since  been  established.* 

In  view  of  the  fact  that  our  Government  has  had,  all  in  all,  nearly 
3,000,000  square  miles  of  land  to  sell  or  to  otherwise  dispose  of,  and 
that  the  sale  had  always  to  be  preceded  by  a  survey,  it  must  be  evident 
that  there  was  a  demand  for  surveyors.  They  could  earn  a  compara- 
tively easy  subsistence  while  a  student  of  pure  mathematics  might  have 
gone  a  begging  for  a  living.  About  1816  a  friend  of  Comte  in  this  coun- 
try warned  that  French  mathematician  and  philosopher  against  the 
purely  practical  spirit  that  prevailed  in  this  new  country,  and  against 
coming  here,  by  saying:  "If  Lagrange  were  to  come  to  the  United 
States  he  could  only  earn  his  livelihood  by  turning  surveyor." 

Mathematical  Journals. 

The  number  of  mathematical  journals  published  in  this  country  since 
the  beginning  of  this  century  is  much  greater  than  one  might  suppose. 
A  full  historical  sketch  of  these  periodicals  has  been  given  by  Dr.  David 
S.  Hart  in  the  Analyst  (Vol.  II,  pp,  131-8, 1875),  and  we  shall  make  free 
use  of  his  valuable  article. 

The  oldest  mathematical  journal  in  America  was  the  Mathematical 
Correspondent.  It  was  established  by  gentlemen  in  I^iTew  York  and  other 
cities,  who  had  long  felt  the  want  of  a  periodical  which  should  do  for 
America  what  the  Ladies'  Diary  had  done  for  England.  George  Baron 
was  editor-in-chief.  It  was  to  be  issued  quarterly.  The  first  number 
was  issued  in  New  York  City  on  May  1, 1804.  Only  eight  numbers  ever 
appeared.  An  essay  in  this  magazine  on  Diophantine  analysis,  by 
Eobert  Adrain,  was  the  first  attempt  to  introduce  the  study  of  this  sub- 
ject in  America. 

The  main  cause  of  the  discontinuance  of  the  journal  lies  in  the  prej- 
udice which  the  editors,  who  were  of  Hibernian  descent,  entertained 
against  American  authors.  A  contributor,  who  called  himself  "  A  Eab- 
bit,"  was  permitted  by  the  editors  to  sneer  at  several  works  written  by 
American  authors,  such  as  Shepherd,  Pike,  Walsh,  and  others.  The 
editors  themselves  also  spoke  in  the  most  contemptuous  manner  of  Col. 
Jared  Mansfield,  the  superintendent  of  the  Military  Academy  at  West 
Point.    Baron  advertised  on  the  cover  of  No.  2  of  the  Correspondent  a 

*  For  fiirtlier  information  on  tLe  early  surveys,  see  Niles's  Register,  Vol.  XII,  pp. 
97,406;  Vol.  XVI,  p.  363. 


INFLUX   OF   ENGLISH  MATHEMATICS.  95 

lecture  delivered  by  him  in  New  York,  which  contains,  as  he  says,  "  a 
complete  refutation  of  the  false  and  spurious  principles  ignorantly  im- 
posed on  the  public  in  the  new  American  Practical  Navigator,  written 
by  N.  Bowditch."  The  sub-editors  endorsed  the  above.  But  some  of 
these  attacks,  especially  "  A  Rabbit's,"  seem  to  have  created  trouble, 
and  on  p.  154  the  editor  says :  '' '  A  Eabbit'  will  not  in  any  future  num- 
ber be  permitted  to  propose  questions  concerning  the  blunders  of  stupid 
Shepherds;  we  had  rather  soar  aloft  with  the  eagle  than  waddle  in  the 
mud  with  the  goose."  For  some  hidden  reason,  Baron  resigned  the 
editorship.  Many  of  the  subscribers  neglected  to  pay,  and  the  paper 
soon  died  out. 

The  next  periodical  was  the  Analyst,  or  xMathematical  Museum,-  edited 
by  Eobert  Adrain.  The  first  number  was  published  in  Philadelphia  in 
1808.  Five  numbers  only  appeared.  We  have  spoken  of  this  periodical 
at  some  length  when  we  wrote  about  Robert  Adrain.  It  contained  the 
valuable  original  work  of  Adrain  on  the  Law  of  Probability  of  Errors. 
Besides  the  editor,  N.  Bowditch,  Alexander  M.  Fisher,  and  Melatiah 
Nash  were  among  the  contributors  to  the  Analyst. 

In  1818,  William  Marrat  became  editor  of  the  Scientific  Journal,  which 
was  published  at  Perth  Amboy,  N.  J,,  in  monthly  numbers.  Nine  num- 
bers are  all  that  are  known  to  have  appeared.  The  cause  of  the  discon- 
tinuance seems  to  have  been  the  departure  of  Mr.  Marrat  for  England. 

In  1825  Eobert  Adrain  started  in  New  York  a  second  periodical,  the 
Mathematical  Diary,  which  was  published  quarterly  during  the  first 
two  years  and  annually  during  the  last  two.  The  last  number  contains 
an  excellent  likeness  of  Lagrange,  and  a  sketch  of  his  life.  After  the 
first  year  the  editorship  of  the  journal  passed  into  the  hands  of  James 
Eyan,  the  author  of  several  mathematical  works.  In  the  preface  to 
the  first  number  of  the  Mathematical  Diary,  Eobert  Adrain  said :  "  The 
principal  object  of  the  present  little  work  is  to  excite  the  genius  and 
industry  of  those  who  have  a  taste  for  mathematical  studies  by  afford- 
ing them  an  opportunity  of  laying  their  speculations  before  the  public 
in  an  advantageous  manner.  *  *  *  It  is  well  known  to  mathema- 
ticians that  nothing  contributes  more  to  the  development  of  mathemati- 
cal genius  than  the  efforts  made  by  the  student  to  discover  the  solu- 
tions of  new  and  interesting  questions."  These  words  may  have  been 
prompted  by  his  own  experience.  We  have  already  pointed  out  how 
the  Analyst,  which  was  edited  by  him  seventeen  years  previously,  was 
the  medium  of  publishing  the  first  proofs  of  the  all  important  Law  of 
the  Facility  of  Error  in  Observations. 

Nearly  all  the  more  prominent  mathematicians  of  America  were  con- 
tributors to  the  Diary.  Among  them  were  Eobert  Adrain,  N.  Bow- 
ditch, Theodore  Strong,  Eugene  Nulty,  Benjamin  Peirce,  Benjamin 
Hallowell,  William  Lenhart,  M.  O'Shannessy,  Henry  J.  Anderson,  and 
others. 

In  1832  the  publication  was  suspended  on  account  of  an  unfortunatQ 


96  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

quarrel  among  the  mathematicians.  Mr.  Samuel  Ward,  a  then  recent 
graduate  of  Columbia  College,  had  in  part  the  management  of  the  last 
number,  in  which  he  inserted  a  dialogue,  written  by  himself,  exhibiting 
in  a  ridiculous  light  Henry  J.  Anderson,  then  professor  of  mathematics 
at  Columbia  College.  High  words  passed  between  the  parties  and  their 
friends,  which  resulted  in  the  complete  breaking  up  of  the  Mathemati- 
cal Diary.  Samuel  Ward  was  afterward  editor  of  Young's  Algebra. 
In  later  years  he  followed  wholly  different  pursuits.  He  became  known 
in  Washington  as  the  "  king  of  the  lobby,"  and  as  the  giver  of  the  best 
dinners  of  any  man  in  America. 

According  to  Dr.  Hart,  a  journal  called  the  Mathematical  Companion, 
was  started  by  John  D.  Williams  in  1828,  and  continued  for  four  years. 
The  periodical,  says  Dr.  Hart,  was  evidently  gotten  up  as  a  rival  of  the 
Mathematical  Diary.  The  writer  has  never  seen  a  copy  of  this  periodi- 
cal. There  is  one  in  the  Harvard  library.*  Mr.  Williams  had  many 
opponents,  and  a  bitter  contest  was  carried  on  between  the  two  parties. 
He  finally  issued  his  fourteen  famous  "  challenge  problems,"  directed 
against  all  the  mathematicians  in  America,  excepting  only  Dr.  Bow- 
ditch,  Professor  Strong,  and  Eugene  Nulty.  Six  of  these  are  impossible. 
All  the  others  have  been  solved  by  several  persons.! 

The  periodicals  which  we  have  named  were  devoted  entirely  to  math, 
ematics.    In  addition  to  these  there  wera  publications  which  were  given 

*  Dr.  Artemas  Martin  sends  us  the  full  title  of  the  journal,  as  found  in  Bolton's 
Catalogue  of  Scientific  and  Technical  Periodicals,  1665  to  1882,  published'  by  the 
Smithsonian  Institution,  p. 360 — "The  Mathematical  Companion,  containing  newre- 
eearches  and  improvements  in  the  mathematics,  with  collections  of  questions  proposed 
and  resolved  by  ingenious  correspondents.  Edited  by  Williams;  1  vol.,  18  mo,,  New 
York,  1828-31." 

tin  the  Educational  Notes  and  Queries,  edited  by  W.  D.  Henkle,  Vol.  II,  No.  11, 
January,  1876,  will  befound  a  copy  of  a  communication  to  a  newspaper  made  by  John 
D.  Williams  in  1832,  containing  the  "  fourteen  challenge  problems,"  and  beginning  as 
follows  ; 

"Messrs.  Editors.— It  is  this  day  six  months  since,  under  the  signature  of  Diophmiim, 
I  proposed  through  the  medium  of  your  paper  to  the  mathematicians  of  America,  a 
collection  of  problems  in  Diophantine  analysis.  No  correct  solution  having  as  yet 
been  received  to  the  whole  of  them,  I  take  this  opportunity  to  fulfill  my  pledge  to 
furnish  such,  and  inclosed  they  will  come  to  your  hands.  I  now  desire  to  re-propose 
them  for  the  ensuing  six  months;  and  shall  except  from  ray  challenge  the  Hon.  Na- 
thaniel Bowditch,  LL.  D.,.  etc.,  of  Boston,  Mass. ;  Mr.  Eugene  Nulty,  of  Philadel- 
phia; and  Prof.  Theodore  Strong,  of  Eatgers  College,  New  Brunswick,  N.  J.,  oiily. 
The  list  of  gentlemen  challenged  stands  then  as  follows:  Prof.  Robert  Adrain, Uni- 
versity of  Pennsylvania;  Henry  J.  Anderson,  Columbia  College,  N.  Y. ;  Benjamin 
Peirce,  Harvard  University,  Cambridge,  Mass. ;  Mr.  J.  Ingersoll  Bowditch,  Boston, 
Mass. ;  Marcus  Catlin,  Hamilton  College,  Clinton,  N.  Y. ;  M.  Floy,  jr., New  York;  C. 
Gill,  Sawpitts  Academy,  N.  Y, ;  L.  L.  Incounew,  Cincinnati,  Ohio  ;  lienjamin  Hallo- 
well,  Alexandria,  Va. ;  Samuel  Ward,  3rd  New  York — it  being  presumed  that  there  are 
none  in  the  United  States  with  the  exception  of  the  above  list  would  think  of  attempt- 
ing their  solution."  Then  follow  the  fourteen  questions.  All  problems  being  in  Dio- 
phantine analysis  would  tend  to  show  that  this  subject  was  then  a  comparatively 
favorite  study. 


INFLUX    OP   ENGLISH   MATHEMATICS.  97 

to  science,  or  to  useful  information  in  general,  but  which  gave  part  of 
their  space  to  a  "  mathematical  department."  Foremost  among  these 
was  the  Ladies  and  Gentlemen's  Diary,  or  United  States  Almanac,  etc., 
edited  by  Melatiah  Nash,  for  the  years  1820,  '21,  '22.  It  contained  much 
valuable  information  in  astronomy  and  philosophy,  enigmas,  charades, 
queries,  and  mathematical  problems,  to  be  answered  in  the  succeeding 
numbers.  Other  almanacs  which  generally  contained  mathematical 
problems  were  Thomas's  Almanac,  published  at  Worcester,  Mass., 
which  existed  for  more  than  one  hundred  years ;  the  Maine  Farmer's 
Almanac ;  two  publications,  each  called  the  "  Farmers'  Almanac ;"  the 
Knickerbocker  Almanac j  the  Anti-Masonic  Almanac,  commenced  in 
1828  at  Eochester,  N.  T.  Other  journals  having  a  mathematical  de- 
partment were  the  American  Monthly  Magazine,  commenced  in  Kew 
York  in  the  year  1817 ;  the  Portico,  which  was  started  in  Baltimore  in 
1816  and  continued  two  or  three  years. 

The  mathematical  journals  spoken  of  were  all  of  the  most  elementary 
kind,  and,  excepting  I^o.  lY  of  the  Analyst,  which  contained  Adrain's 
investigations  on  least  squares,  added  nothing  to  the  stock  of  mathe- 
matical science.  These  journals  had  an  educational  rather  than  scien- 
tific value.  The  proposal  and  solution  of  problems  was  the  main  work 
done  by  their  contributors,  ^ow,  it  will  certainly  be  admitted  that 
solving  problems  is  one  of  the  lowest  forms  of  mathematical  work.  The 
existence  of  mathematical  journals  shows  that  since  the  beginning  of 
this  century  there  always  were  some  persons  interested  in  mathematics, 
but  the  number  was  so  small  that  mathematical  journals  never  were 
a  financial  success.  All  the  early  mathematical  periodicals  had  merely 
an  ephemeral  existence. 
881—^0.  3 7 


III. 


THE  INFLUX  OF  FEENOH  MATHEMATICS. 

During  the  latter  part  of  the  eighteenth  century  we  see  the  French 
people  rising  with  fearful  unanimity,  destroying  their  old  institutions, 
and  upon  their  ruins  planting  a  new  order  of  things.  With  this  period 
begins  the  interest  in  popular  education  in  France.  A  new  impetus 
was  given  also  to  higher  scientific  education,  which  continued  to  be  far 
in  advance  of  that  of  the  rest  of  Europe. 

In  1794  was  opened  in  Paris  the  Polytechnic  School  and  in  the  fol- 
lowing year  the  Schools  of  Application.  The  Polytechnic  School 
gained  a  world-wide  celebrity.  The  professors  at  this  institution  were 
men  whose  names  are  household  words  wherever  science  has  a  votary. 
Lagrange,  Lacroix,  and  Poisson  laid  the  basis  to  its  course  in  analytical 
mathematics;  Laplace,  Ampere,  and  others  to  that  of  analytical  me- 
chanics and  astronomy.  Descriptive  geometry  and  its  applications  had 
for  their  first  teachers  the  founder  of  this  science,  the  illustrious  Monge 
and  his  celebrated  pupils,  Hachette  and  Arago. 

The  success  of  the  Polytechnic  School  was  phenomenal.  It  was  the 
nurse  of  giants.  Among  its  pupils  were  Arago,  Biot,  Bourdon,  Cauchy, 
Chasles,  Duhamel,  Dupin,  Gay-Lussac,  Le  Verrier,  Poncelet,  Eegnault. 
The  Polytechnic  School  is  of  special  interest  to  those  who  live  in  America, 
because  the  U.  S.  Military  Academy  at  West  Point  was  a  germ  from  it. 

Compared  with  the  French  mathematicians  who  flourished  at  the 
beginning  of  this  century  the  contemporary  American  professors  were 
mere  Liliputians.  The  masterpieces  of  French  scholars  were  unknown 
in  America.  What  little  mathematical  knowledge  existed  here  came 
to  us  through  English  channels.  For  that  reason  that  epoch  was  called 
the  period  of  the  influx  of  English  mathematics.  As  compared  with 
colonial  tiuies,  considerable  attention  was  paid  to  mathematical  studies 
during  that  period.  But  there  was  still  a  great  dearth  in  original 
thinkers  on  mathematics  among  us.  The  genius  of  our  people  was  ex- 
ercised  in  different  .fields,  and  so  the  little  science  we  had  was  borrowed 
from  others. 

But  the  time  came  when  French  writers  were  at  last  beginning  to 

make  their  influence  felt  among  us.     We  recognized  their  superiority 

over  the  English  and  profited  by  it.    Mathematical  studies  received  a 

new  impetus.    But  even  then  ours  was  not  the  glory  of  the  sun,  but 

98 


INFLUX   OF  FRENCH   MATHEMATICS.  99 

only  of  the  moon.  The  new  period  produced  among  us  only  one  mathe- 
matician displaying  real  genius  for  original  research. 

It  is  naturally  humiliating  to  an  American  when  a  foreign  mathema- 
tician like  Todhunter,  well  known  for  the  fairness  and  candor  of  his 
views,  pronounces  a  judgment  on  Americans  like  the  following:  "I 
have  no  wish  to  depreciate  their  labors  ;  I  know  that  they  possess  able 
mathematicians,  and  that  in  the  department  of  astronomy  they  have 
produced  meritorious  works ;  but  I  maintain  that  as  against  us  their 
utmost  distinction  almost  vanishes.  And  yet,  with  their  great  popula- 
tion, their  abundant  wealth,  their  attention  to  education,  their  freedom 
from  civil  and  religious  disabilities,  and  their  success  in  literature,  we 
might  well  expect  the  most  conspicuous  eminence  in  mathematics."* 

No  thinking  American  will  pronounce  this  estimate  of  American 
mathematicians  as  entirely  unsound ;  it  is,  in  fact,  quite  correct.  The 
reasons  for  this  want  of  productiveness  certainly  do  not  lie  in  any  lack 
of  power  in  the  American  mind.  They  will  be  found  rather  (1)  in  the 
want  of  interest  in  and  appreciation  of  abstract  scientific  work  on  the 
part  of  the  American  people,  and  (2)  in  the  bad  methods  of  mathemat- 
ical instruction  in  our  elementary  and  higher  institutions  of  learning. 
There  has  been  no  incentive  in  this  country  for  any  large  body  of  men 
to  direct  their  life-work,  day  by  day,  in  the  line  of  mathematical  inves- 
tigation. In  former  years  our  professors  in  colleges  were,  with  few 
exceptions,  over- worked  in  the  recitation  room;  their  routine  work 
absorbed  all  their  energies,  thereby  rendering  their  minds  unfit  for 
original  research.  Again,  every  teacher  had  a  stomach;  his  wife  and 
children  had  stomachs;  the  human  being  must  be  fed;  a  livelihood 
must  be  earned ;  the  professor's  salary  was  low ;  not  unfrequently  he 
had  to  add  to  his  duties  as  instructor  in  college  those  of  a  preacher  or 
private  teacher,  in  order  to  make  his  living.  Such  conditions  were  not 
favorable  for  the  growth  of  science. 

But,  in  spite  of  all  difficulties,  there  was  much  progress.  The  im- 
provements in  mathematical  text-books  and  reforms  in  mathematical  in- 
struction were  due  to  French  influences.  French  authors  displaced  the 
English  in  many  of  our  best  institutions.  It  is  somewhat  of  a  misfor- 
tune, however,  that  we  failed  to  gather  in  the  full  fruits  of  the  French 
intellect.  We  followed  in  the  path  of  French  writers  whose  works  had 
ceased  to  be  the  embodiment  of  the  later  results  of  French  science ; 
many  of  the  works  which  we  adopted  were  beginning  to  h6  "  behind 
the  times,"  when  introduced  in  America.  We  used  works  of  Bezout, 
Lacroix,  and  Bourdon.  But  Bezout  flourished  before  the  French  Rev- 
olution, and  Lacroix  wrote  most,  if  not  all,  of  his  books  before  the  be- 
ginning of  this  century.  In  182L  Cauchy  published  in  Paris  his  Gours 
(^Analyse.  If  thoughtful  attention  and  study  had  been  given  by  our 
American  text-book  writers  to  this  volume,  then  many  a  lax,  loose,  and 
unscientific  method  of  treating  mathematical  subjects  might  have  been 

•  Tke  Conflict  of  Studies  aad  Other  Essays,  by  I.  Todhuuter.    London,  1873,  p.  160. 


100  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

corrected  at  the  outset.  The  wretched  treatment  of  infinite  series,  as 
found  in  all  our  text-books,  excepting  the  most  recent,  might  have  been 
rejected  from  the  very  beginning. 

In  thinking  of  the  influx  of  French  mathematics,  we  must  guard 
against  the  impression  that  French  authors  and  methods  entirely  dis- 
placed the  English.  English  books  continued  to  be  used  in  some  of 
our  schools.  Many  an  old  English  notion  has  remained  with  us  to  the 
present  day.  We  still  have  the  English  weights  and  measures.  The 
old  line-system  in  trigonometry,  which  we  got  from  tlue  English,  but 
which  they  long  since  rejected,  has  until  very  recently  been  finding 
favor  among  many  of  our  teachers. 

There  have  been  improvements  in  the  methods  of  instruction,  but 
not  so  extensive  as  might  be  wished.  Traditional  methods  have  long 
had  almost  full  sway.  The  mathematical  teaching  has  been  bad.  One 
of  the  most  baneful  delusions  by  which  the  minds,  not  only  of  students, 
but  even  of  many  teachers  of  mathematics  in  our  classical  colleges, 
have  been  afflicted  is,  that  mathematics  can  be  mastered  by  the  favored 
few,  but  lies  beyond  the  grasp  and  power  of  the  ordinary  mind.  This 
chimera  has  worked  an  untold  amount  of  mischief  in  mathematical  edu- 
cation. The  students  entered  upon  their  studies  with  the  feeling  that 
there  was  no  use  trying  to  learn  mathematics,  and  the  teacher  felt  that 
there  was  no  use  trying  to  teach  it.  This  humiliating  opinion  of  the 
powers  of  the  average  human  mind  is  one  of  the  most  unfortunate 
delusions  which  have  ever  misled  the  minds  of  American  students  and 
educators.  It  has  prevailed  among  us  from  the  earliest  times.  In  the 
latter  part  of  the  last  century,  the  notion  was  general  among  us  that 
girls  could  not  be  taught  fractions  in  arithmetic,  and  that  lady  teachers 
were  unfit,  for  want  of  mental  capacity,  to  give  instruction  in  arithme- 
tic. Warren  Burton  says  that  a  school-mistress  "  would  as  soon  have 
expected  to  teach  the  Arabic  language  as  the  numerical  science."  But 
this  delusion  has  now  vanished.  The  best  instruction  in  elementary 
arithmetic  is  now  given  by  lady  teachers.  Among  the  contributors  to 
the  American  Journal  of  Mathematics  there  are  two  ladies.  In  the 
same  way  the  delusion  will  soon  vanish  that  the  average  college  stu- 
dent is  not  able  to  grasp  the  more  advanced  branches  of  exact  science. 
The  trouble  has  been,  all  alon^,  not  so  much  in  the  lack  of  ability  in 
students,  as  in  the  wretched  character  of  the  mathematical  instruction. 
Such  is  the  opinion  of  Professor  Olney,  one  of  the  most  efficient  drill- 
masters  and  teachers  of  mathematics  that  this  country  has  produced. 
In  the  preface  to  his  General  Geometry  and  Calculus  he  says :  "  Nor  is 
it  impracticable  for  the  majority  of  students  to  become  intelligent  in 
these  subjects.  They  do  not  lie  beyond  the  reach  of  good  common  minds, 
nor  require  peculiar  mental  characteristics  for  their  mastery.  The  dif- 
ficulty hitherto  has  been  in  the  methods  of  presentation,  in  the  limited 
and  totally  inadequate  amount  of  time  assigned  them,  and  more  than 
all  in  the  preconceived  notion  of  their  abstruseuess." 


INFLUX  OF  FRENCH  MATHEMATICS.  101 

One  of  the  causes  of  the  bad  instruction  in  our  colleges  has  been  the 
system  of  tutorships.  Fortunately,  this  relic  of  scholasticism  is  now 
rapidly  disappearing.  Young  students  "who  needed  a  skilled  teacher  of 
long  experience  to  guide  them  and  to  awaken  in  them  a  spirit  of  free 
inquiry  were  intrusted  to  inexperienced  youths  who  had  just  gradu- 
ated from  college,  and  who  had  themselves  never  felt  the  glow  of  the 
spirit  of  independent  inquiry.  Students  did  not  find  their  mathematics 
interesting,  nor  did  they  understand  it  well.  Their  hatred  of  mathe- 
matics had  its  cause  in  these  two  facts,  which  stand  in  the  closest  pos- 
sible connection  with  each  other.  ''  We  might  say,  either  that  the  study 
failed  of  being  understood,  because  it  was  uninteresting,  or  that  it 
awakened  no  interest,  because  it  was  not  well  understood.  Both  these 
statements  were  true."*  Professor  Eddy  truly  says  that  very  few  stu- 
dents "  do  really  become  in  any  true  sense  masters  of  the  mathematical 
subjects  which  they  study,  or  indeed  have  sufficient  practice  in  the 
principles  which  they  attempt  to  learn,  to  be  capable  of  judging  whether 
they  have  been  so  mastered  as  to  accomi^lish  the  ends  which  should  be 
sought  in  mathematical  training."  The  great  desideratum  in  our  pre- 
paratory schools  and  colleges  has  been  less  memorizing,  less  csamming, 
more  thorough  training  in  the  fundamental  branches,  more  obiect  teach- 
ing, more  drill,  more  frequent  and  well-guided  original  inquiries,  greater 
freedom  from  formalism,  a  stronger  spirit  of  free  inquiry. 

,Says  Professor  Eddy :  "  When,  as  often  happens,  our  college  grad- 
uates go  abroad  for  post-graduate  study  in  departments  requiring  pre- 
vious mathematical  traiaing,  what  do  they  fiad  their  requirements  in 
this  direction  to  amount  to  ?  I  think  I  may  say  that  a  large  proportion 
of  them  find  themselves  almost  hopelessly  lacking  in  the  essentials  of 
such  training,  and  not  at  all  fitted  to  make  proper  improvement  of  the 
advantages  of  which  they  have  sought  to  avail  themselves.  Our  young 
men  are  unequal  to  the  mathematical  studies  which  those  of  the  same 
age,  but  of  European  academic  training,  successfully  carry.  Now, 
where  does  the  difficulty  lie?  Kot  in.  any  inferior  talent  for  these 
studies,  as  I  have  the  best  of  reasons  for  believing,  but  from  a  lack  of 
opportunity  for  obtaining  a  comprehension  of  the  infinitesimal  calculus, 
in  which  they  usually  find  themselves  almost  wholly  wanting."  Nor 
are  they  always  able  to  manipulate,  with  any  degree  of  ease,  the  more 
complicated  expressions  of  ordinary  algebra.  They  have  been  taught 
by  a  "  daily  lecture  instead  of  a  daily  drill,"  a  method  of  teaching  which 
is  like  "  explaining  tactics  instead  of  practicing  them."  Or,  whenever 
text-books  were  used,  ''  the  recitations  were  mere  hearings  of  lessons, 
without  comment  or  collateral  instruction."! 

Professor  .Eddy's  reminiscences  of  his  own  study  of  mathematics  in 
college  are  not  ijleasant.    E"or  is  his  experience  exceptional.    On  the  con- 

*  "  College  Mathematics,"  by  Henry  T.  Eddy,  iu  the  Proceedings  of  the  Amer- 
iciin  Association  for  the  Advancement  of  Science,  Vol.  XXXIII,  1884. 
t  Harvard  Keminisceuces,  by  A.  P.  Peabody,  p.  201. 


102  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

trary  it  has  been  the  rule  rather  than  the  exception  in  our  classical  col- 
leges. In  reply  to  a  request  made  by  the  'writer  to  give  his  recollections 
of  the  mathematical  teaching  at  one  of  our  oldest  classical  colleges,  a 
now  prominent  professor  of  mathematics  replied  that  he  did  not  think 
he  had  "  aoy  such  recollections"  as  he  "  should  care  to  put  in  print." 
Another  one  gives  his  reminiscences,  but  marks  his  letter  "personal  and 
private." 

If  our  classical  colleges  had  caught  something  of  the  spirit  that  must 
have  prevailed  at  the  Polytechnic  School  in  Paris  in  the  days  of  La- 
grange, Laplace,  Lacroix,  Ampere,  when  it  produced  such  thinkers  as 
Arago,  Cauchy,  Le  Verrier,  then  the  list  of  our  prominent  mathemati- 
cians and  astronomers  would  doubtless  have  been  doubled  or  tripled. 
We  got  from  the  French  some  of  their  old  text-books,  but  we  failed  to 
catch  their  love  of  scientific  study  and  inquiry. 

On  a  previous  page  it  has  been  stated  that  Americans  had  come  to 
recognize  the  superiority  of  French  mathematicians  over  the  English. 
It  should  have  been  added  that  we  did  not  see  this  superiority  until  it 
was  pointed  out  to  us  by  the  English  themselves.  The  influx  of  French 
mathemaj;ics  into  the  United  States  was  preceded  by  an  influx  of  French 
mathematics  into  England.  In  Britain  there  were  men  who  had  come 
to  deplore  the  very  small  progress  that  science  was  making  there,  as 
compared  with  its  racing  progress  on  the  continont.  In  1813  the 
"Analytical  Society"  was  formed  at  Cambridge.  This  was  a  small 
club  established  by  Peacock,  John  Herschel,  Babbage,  and  a  few  other 
students  at  Cambridge,  to  promote,  as  it  was  humorously  expressed,  the 
principles  of  pure  "  D-ism,"  that  is,  of  the  Leibnitzian  notation  in  the 
calculus,  against  those  of  "  dot-age,"  or  of  the  Newtonian  notation.  This 
struggle  ended  in  the  introduction  into  Cambridge  of  the  Continental  no- 
tation (^)  to  the  exclusion  of  the  fluxional  notation  {y).    This  was  a 

great  step  in  advance,  not  on  account  of  any  great  superiority  of  the 
Leibnitzian  over  the  Newtonian  notation,  but  because  the  adoption  of 
the  former  opened  up  to  English  students  the  vast  storehouses  of  Con- 
tinental discoveries. 

The  movement  against  the  fluxional  notation  began  in  this  country 
almost  ten  years  later  than  it  did  in  England,  and  i^roceeded  more 
quietly.  John  Farrar,  of  Harvard,  translated  from  the  French  the  Dif- 
ferential and  Integral  Calculus  of  Bezout,  which  employed  the  Continental 
notation,  in  1824.  Professor  Fisher,  of  Yale,  who  died  in  1822,  published 
mathematical  articles  in  Silliman's  Journal,  employing  the  new  nota- 
tion. At  an  earlier  date  than  this  there  were  men  connected  with  West 
Point  who  had  been  trained  in  the  Continental  system.  Thus,  F.  E. 
Hassler,  educated  at  the  University  of  Bern,  was  teacher  Of  mathemat- 
ics at  West  Point  from  1808  to  1810.  Probably  neither  calculus  nor 
fluxions  were  taught  there  during  that  time,  for,  as  late  as  181G,  we  read 
in  the  West  Point  curriculum  thdit  jiuxions  were  "  to  be  taught  at  the 


INFLUX  OF  FEENCH  MATHEMATICS.  103 

option  of  professor  and  student."  In  1817,  Orozet,  trained  at  the  Poly- 
technic School  in  Paris,  became  teacher  of  engineering  at  the  Military 
Academy.  In  this  country  he,  sometimes  at  least,  used  the  jJ^ewtonian 
notation.  He  did  so,  for  instance,  in  the  solution,  in  French,  of  a  prob- 
lem which  he  published  in  the  Portico,  of  Baltimore,  in  1817.  The 
Leibnitzian  notation  must  have  been  introduced  at  the  Military  Acad- 
emy very  soon  after  the  year  1817. 

Eobert  Adrain  used  the  English  notation  in  his  earlier  writings.  In 
the  Portico,  Vol.  Ill,  he  does  so,  but  in  Nash's  Ladies  and  Gentlemen's 
Diary,  No.  II,  published  in  New  York  in  1820,  he  employs  the  notation 
dx.  We  are  told  that  while  he  was  at  Columbia  College,  between  1813 
and  1826,  he  wrote  a  manuscript  treatise  on  the  Differential  and  Integral 
Calculus.  We  know  also  that  he  was  a  diligent  student  of  the  works 
of  Lagrange  and  Laplace,  which  contained  the  notation  of  Leibnitz 
throughout.  The  first  article  in  the  Memoirs  of  the  American  Acad- 
emy of  Arts  and  Sciences,  which  contains  the  "d-istic"  notation,  was 
published  in  1818  by  F.  T.  Schubert.  It  is  well  known  that  Bowditch 
began  the  translation  of  the  Mecanique  Celeste  of  Laplace  as  early  as 
1814.  At  that  time  he  was,  therefore,  thoroughly  conversant  with 
pure  "D-ism."  He  had  been  converted  to  the  new  "ism"  on  the  long 
sea  voyages,  from  1795  to  180i,  when  he  studied  Lacroix's  Calculus. 
In  general,  it  maybe  stated  that  the  change  of  notation  took  place  in 
the  United  States  about  the  close  of  the  first  quarter  of  this  century. 

The  publication  of  Bowditch's  Laplace,  begun  in  1829,  gave  a  pow- 
erful stimulus  to  the  study  of  French  mathematics  and  to  the  general 
advancement  of  mathematical  learning  in  America.  Says  Edward 
Everett :  "  This  may  be  considered  as  opening  a  new  era  in  the  history 
of  American  science." 

This  may  be  a  convenient  place  to  consider  that  work  at  some  length. 
As  it  originally  appeared  in  France,  the  M6canique  Celeste  was  de- 
scribed by  the  Edinburgh  Review,  one  of  the  kadiug  scientific  journals 
in  Great  Britain,  as  being  of  so  abstruse  and  profound  a  character  that 
there  were  scarcely  a  dozen  men  in  all  that  country  capable  of  reading 
it  with  any  tolerable  facility.  These  remarks  created  great  curiosity 
in  Bowditch  to  explore  the  work.  He  began  translating  it  in  1814,  and 
pursued  it  with  such  ardor  and  persistence  that  he  accomplished  it  in 
only  two  years. 

In  order  to  state  briefly  the  object  of  the  work  of  La  Place,  we  quote 
from  his  preface  to  it  as  follows : 

"Toward  the  end  of  the  seventeenth  century,  Newton  published  his 
discovery  of  universaV  gravitation.  Mathematicians  have  since  that 
epoch  succeeded  in  reducing  to  this  great  law  of  nature  all  the  knojvn 
phenomena  of  the  system  of  the  world,  and  have  thus  given  to  the  theo- 
ries of  the  heavenly  bodies  and  to  astronomical  tables  an  unexpected 
degree-of  precision.  My  object  is  to  present  a  connected  view  of  these 
theories  which  are  now  scattered  in  a  great  number  of  works.  The 
whole  of  the  results  of  gravitation  upon  the  equilibrium  and  motions 


104  TEACHING   AND    HISTORY   OF   MATHEMATICS. 

of  the  fluid  aud  solid  bodies  whicli  compose  tlie  solar  system  and  tha 
similar  systems  existing*  in  the  immensity  of  space,  constitute  the  object 
of  Celestial  Mechanics,  or  the  application  of  the  j)rinciples  of  mechan- 
ics to  the  motions  and  figures  of  the  heavenly  bodies.  Astronomy,  con- 
sidered in  the  most  general  manner,  is  a  great  problem  of  mechanics, 
in  which  the  elements  of  the  motions  are  the  arbitrary  constant  quanti- 
ties. The  solution  of  this  problem  depends,  at  the  same  time,  upon  the 
accuracy  of  the  observations  and  upon  the  perfection  of  the  analysis. 
It  is  very  important  to  reject  every  empirical  process,  and  to  complete 
the  analysis,  so  that  it  shall  not  be  necessary  to  derive  from  observa- 
tions any  but  indispensable  data.  The  intention  of  this  work  is  to 
obtain,  as  much  as  may  be  in  my  i30wer,  this  interesting  result." 

Though  the  translation  was  completed  as  early  as  1817,  the  publica- 
tion did  not  begin  until  1829.  In  1817  the  income  of  Bowditch  was  so 
small  that  he  could  not  afford  to  have  the  translation  published.  The 
American  Academy  of  Arts  and  Sciences  offered  to  publish  the  work  at 
their  own  expense.  He  was  also  solicited  to  publish  it  by  subscription. 
But  his  independence  of  spirit  induced  him  to  decline  these  proposals. 
He  was  aware  that  the  work  would  find  but  few  readers,  and  he  did  not 
wish  any  one  to  feel  compelled  or  to  be  induced  to  subscribe  for  it,  lest 
he  should  have  it  in  his  power  to  say,  "  I  patronized  Mr.  Bowditch  by 
buying  his  book,  which  I  can  not  read."  Later  on  he  was  able  to  com- 
mence the  publication  at  his  own  expense. 

The  objects  which  Bowditch  endeavored  to  accomplish  by  his  trans- 
lation and  commentary,  as  stated  by  his  biographers,  were  as  follows: 

(1)  To  supply  those  steps  in  the  demonstration  which  could  not  be 
discovered  without  much  study,  and  which  had  rendered  the  original 
work  so  difficult.  The  difficulty  arose  not  merely  from  the  intrinsic  com- 
plexity of  the  subject  and  the  medium  of  proof  by  the  higher  branches 
of  mathematics,  but  chiefly  from  the  circumstance  that  the  author, 
taking  it  for  granted  that  the  subject  would  be  as  plain  and  easy  to 
others  as  to  himself,  very  often  omits  the  intermediate  steps  and  con- 
necting links  in  his  demonstrations.  He  jumps  over  the  interval  and 
grasps  the  conclusion  by  intuition.  Bowditch  used  to  say,  "  I  never 
come  across  one  of  Laplace's  '  Thus  it  plainly  appears^  without  feeling 
sure  that  I  have  hours  of  hard  work  before  me  to  fill  up  the  chasm  and 
find  out  and  show  how  it  plainly  appears."* 

(2)  The  second  great  object  of  the  translation  was  to  continue  the 
original  work  to  the  present  time,  so  as  to  include  the  many  improve- 
ments and  discoveries  in  mathematical  science  that  had  been  made 
during  the  twenty-five  years  succeeding  the  first  publication.  It  is 
gratifying  to  know  that  the  most  eminent  of  contemporary  mathemati- 

*  "The  M6caniqne  C61est6  is  by  no  means  easy  reading,  Biot,  who  assisted  La- 
place in  revising  it  for  the  press,  says  that  Laplace  himself  was  frequently  unable 
to  recover  the  details  in  the  chain  of  reasoning,  aud  if  satisiied  that  the  conclusions 
■were  correct  he  was  content  to  insert  the  constantly  recurring  formula,  'II  est  ais^  ki 
voir.' "    W.  W.  E,  Ball's  Short  History  of  Mathematics,  p.  387. 


INFLUX  OF  FRENCH  MATHEMATICS.     *       105 

cians  pronounced  his  commentary  a  success,  and  agreed  that  Bowditch 
had  attained  the  end  he  had  in  view,  namely,  to  bring  the  work  up  with 
the  times.  Says  Lacroix,  July  S,  183G :  '•  I  am  more  and  more  aston- 
ished at  a  task  so  laborious  and  extensive.  I  perceive  that  you  do  not 
confine  yourself  to  the  mere  text  of  your  author  and  to  the  elucidations 
which  it  requires,  but  you  subjoin  the  parallel  passages  and  subse- 
quent remarks  of  those  geometers  who  have  treated  of  the  same  sub- 
jects ;  so  that  your  work  will  embrace  the  actual  state  of  the  science  at 
the  time  of  its  publication."  Legendre,  July  2, 1832,  says :  "  Your  work 
is  not  merely  a  translation  with  a  commentary  5  I  regard  it  as  a  new 
edition,  augmented  and  improved,  and  such  a  one  as  might  have  come 
from  the  hands  of  the  author  himself,  if  he  had  consulted  his  true  inter- 
est, that  is,  if  he  had  been  solicitously  studious  of  being  clear."  Mr. 
Babbage,  of  England,  August  5,  1832,  says :  "  It  is  a  proud  circum- 
stance for  America,  that  she  has  preceded  her  parent  country  in  such 
an  undertaking ;  and  we  in  England  must  be  content  that  our  language 
is  made  the  vehicle  of  the  sublimest  portion  of  human  knowledge,  and 
be  grateful  to  you  for  rendering  it  more  accessible."  Similar  testimony 
was  given  by  Bessel  and  Bncke  in  Germany  ;  Puissant  in  France;  Sir 
John  Herschel,  Airy,  Francis  Baily  in  England,  and  Cacciatore  in  Italy. 

Bowditch  once  remarked  that  however  flattering  the  testimony  from 
foreigners  might  be,  yet  the  most  grateful  tribute  of  commendation  he 
had  ever  received  was  contained  in  a  letter  from  a  backwoodsman  of 
the  West,  who  wrote  to  him  to  point  out  an  error  in  his  translation  of 
the  Mecanique  Celeste.  "It  is  an  actual  error,"  said  he,  "which  had 
escaped  my  own  observation.  The  simple  fact  that  my  work  had  reached 
the  hands  of  one  on  the  outer  verge  of  civilization  who  could  under- 
stand and  estimate  it  was  more  gratifying  to  my  feelings  than  the 
eulogies  of  men  of  science  and  the  commendatory  votes  of  Academies." 
In  America,  many  college  professors  were  enabled  by  means  of  the  trans- 
lation and  commentary  to  read  and  understand  the  Mecanique  Celeste, 
who  would  otherwise  have  looked  upon  this  work  as  a  sealed  book. 

Daring  the  first  thirty-five  or  forty  years  of  this  century  but  little  was 
accomplished  in  this  country  in  the  line  of  astronomical  observations. 
More  was  done  in  that  respect  during  the  days  of  David  Eitteuhouse 
than  in  the  early  part  of  this  century.  But,  all  at  once,  a  great  impetus 
was  given  to  this  kind  of  scientific  work.  In  1830  was  erected  the  Yale 
College  Observatory ;  in  1831  the  observ^atory  at  the  University  of  North 
Carolina ;  in  1836  the  Williams  College  Observatory ;  in  1838  the  Hud- 
son Observatory,  Ohio;  in  1840  the  Philadelphia  High  School  Observ- 
atory and  the  West  Point  Observatory;  in  1842  the  IsTational  Observa- 
tory at  Washington.  Since  then  a  large  number  of  other  observatories 
with  excellent  instruments  have  been  built. 

A  plan  for  a  National  Observatory  was  submitted  to  the  G-overnment 
by  Mr.  Hassler,  in  his  project  for  the  survey  of  the  Atlantic  coast,  as 
early  as  1807.    The  proposition  met  with  no  favor.    For  many  years 


106  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

Congress  opposed  every  such  scheme.  John  Quincy  Adams,  in  his 
annual  message  of  1825,  strongly  urged  this  subject  upon  the  atten- 
tion of  Congress.  In  one  place  he  said,  "  It  is  with  no  feeling  of  pride, 
as  an  American,  that  the  remark  may  be  made  that,  on  the  compara- 
tively small  territorial  surface  of  Europe  there  are  existing  upward 
of  one  hundred  a,nd  thirty  of  these  light  houses  of  the  skies ;  while 
throughout  the  whole  American  hemisphere  there  is  not  one."  Presi- 
dent Adams's  appeal  was  received  with  a  general  torrent  of  ridicule. 
"The  proposition,"  says  Loomis,*  <'to  establish  a  light-house  in  the 
skies  became  a  common  by- word  of  reproach."  It  was  not  till  1842 
that  an  appropriation  was  passed  for  an  observatory,  under  the  disguised 
name  of  a  "  Depot  of  Charts  and  Instruments." 

It  need  hardly  be  said  that  in  later  years  theU.  S.  Government  has 
been  very  liberal  in  the  encouragement  of  science. 

Elementary  Schools. 

The  beginning  of  this  period  is  marked  by  a  great  revival  of  element- 
ary education.  Pestalozzian  ideas  had  gained  a  foothold  in  England, 
and  were  now  commencing  to  force  their  way  into  the  western  conti- 
nent. In  1806  F,  J.  N.  Neef,  once  an  assistant  to  Pestalozzi,  came  to 
this  country,  and  began  teaching  and  disseminating  the  ideas  of  the 
Swiss  reformer.  The  first  fruit  of  Pestalozzian  ideas  in  the  teaching 
of  arithmetic  among  us  was  Warren  Colburn's  Intellectual  Arithmetic 
upon  the  Inductive  Method  of  Instruction,  known  as  the  "  First  Les- 
sons." 

Warren  Colburn  worked,  while  a  boy,  at  the  machinist's  trade. 
He  then  entered  Harvard  and  graduated  in  1820,  having  "  mastered 
calculus  and  read  a  large  part  of  Laplace."  He  then  taught  a  select 
school  in  Boston.  At  this  time  he  began  preparing  bis  little  book.  Of 
special  interest  is  the  following  statement  of  Mr.  Batchelder,  of  Cam- 
bridge, which  shows  how  tiie  First  Lessons  were  prepared :  "  I  remember 
once,  in  conversing  with  him  with  respect  to  his  arithmetic,  he  remarked 
that  the  pupils  who  were  under  his  tuition  made  his  arithmetic  for  him  j 
that  he  had  only  to  give  attention  to  the  questions  they  asked  and  the 
proper  answers  and  explanations  to  be  given,  in  order  to  anticipate  the 
doubts  and  difficulties  that  would  arise  in  the  minds  of  the  pupils."  He 
had  read  Pestalozzi,  most  probably,  while  in  college.  A  manuscript 
copy  of  his  First  Lessons  was  furnished  by  Colburn  to  his  friend  George 
B.  Emerson  for  use  in  a  school  for  girls,  and  the  former  received  valua- 
ble suggestions  from  the  latter.  The  success  of  the  book  was  almost 
immediate.  No  school-book  had  ever  had  such  sale  among  us  as  this. 
Over  three  and  one-half  million  copies  were  used  in  this  country,  and  it 
was  translated  into  several  European  languages. 

Colburn's  First  Lessons  embodied  what  was  then  a  new  idea  among  us. 
Instead  of  introducing  the  young  pupil  to  the  science  of  numbers,  as  did 

*  Recent  Progress  in  Astronomy,  OBpeoially  in  the  United  States,  by  Elias  Loomis. 
New  York,  1856,  p.  205. 


INFLUX  OP  FRENCH  MATHEMATICS.  107 

old  Dilworth,  by  the  question,  "  What  is  arithmetic'?"  and  the  answer, 
"Arithmetic  is  the  art  or  science  of  computing  by  numbers,  either  whole 
or  in  fractions,"  he  was  initiated  into  this  science  by  the  following  sim- 
ple question :  "  How  many  thumbs  have  you  on  your  right  hand "?  How 
many  on  your  left  ?  How  many  on  both  together  ?  "  The  idea  was  to 
begin  with  the  concrete  and  known,  instead  of  the  abstract  and  unknown, 
and  then  to  proceed  gradually  and  by  successive  steps  to  subjects  more 
difficult.  In  the  publication  of  this  book,  the  study  of  arithmetic  in  the 
schools  of  this  country  received  its  best  impulse.  "  It  led  to  the  adoption 
of  methods  of  teaching  that  have  lifted  the  mind  from  the  slavery  of 
dull  routine  to  the  freedom  of  independent  thought."    (Edward  Brooks.) 

Colburn's  First  Lessons  was  followed  in  182C  by  his  Arithmetic  upon 
the  Inductive  Method  of  Instruction,  being  a  Sequel  to  Intellectual 
Arithmetic.  This  was  considered  by  its  author  to  be  superior  to  the 
First  Lessons,  but  it  did  not  meet  with  so  great  success.  In  1825  he  pub- 
lished his  Algebra  upon  the  Inductive  Method  of  Instruction.  Mr.  Col- 
burn  did  not  loDg  engage  in  teaching.  Three  years  after  graduation  from 
college  he  was  appointed  superintendent  of  a  manufacturing  company 
at  Waltham,  and,  soon  after,  of  one  at  Lowell,  Mass.  He  possessed 
great  mechanical  genius  and  administrative  ability. 

Though  the  First  Lessons  met  with  ready  appreciation  in  New  Eng- 
land, it  must  not  be  imagined  that  there  was  no  opposition  to  it.  Old 
notions  could  not  be  laid  aside  at  once,  and  even  where  the  new  ideas 
had  gained  entrance,  new  books  could  not  always  be  had  readily. 
Now-a  days  we  are  apt  to  forget  the  difficulty  and  expense  of  trans- 
portation during  the  times  preceding  our  railroad  era.  Says  J.  Stock- 
ton, in  the  i^reface  to  his  Western  Calculator  (fourth  edition,  1823, 
Pittsburg,  Pa.),  "  to  furnish  our  numerous  schools,  in  the  western  (!) 
country,  with  a  plain  and  practical  treatise  of  arithmetic,  compiled  and 
printed  among  ourselves,  thereby  saving  a  heavy  annual  expense  in 
the  purchase  of  such  books,  east  of  the  mountains,  and  likewise  the  car- 
riage thereof,  have  been  the  motives  which  induced  the  compiler  to 
undertake  this  work." 

In  spite  of  all  obstacles  Colburn's  books  gained  ground  steadily. 
Other  books  were  written  upon  the  same  idea  by  different  teachers.  Old 
books  underwent  revision,  so  as  to  embody  the  new  methods  in  part. 
Thus,  the  celebrated  School-masters'  Arithmetic  of  Daniel  Adams,  first 
published  in  UOl,  was  made  to  undergo  a  radical  change.  '  The  old  work 
was  "  synthetic."  "If  that  be  a  fault  of  the  work,"  says  the  author, 
"  it  is  a  fault  of  the  times  in  which  it  appeared.  The  analytic  or  induc- 
tive method  of  Pestalozzi  *  *  *  is  among  the  improvements  of  later 
years.  It  has  been  applied  to  arithmetic  with  great  ingenuity  by  Mr. 
Colburn  in  our  own  country."  "  Instructors  of  academies  and  common 
schools  have  been  so  long  attached  to  the  old  synthetic  method  of  in- 
struction, that,  unhappily,  many  are  still  (1829)  strongly  opposed  to 
the  introduction  of  the  valuable  works  of  Colburn."  "This  [Adams's] 
work  combines  the  new  and  the  old.'' 


108  TEACHING   AND    HISTOKY    OF   MATHEMATICS. 

The  great  success  of  Colburn's  book  did  not  prevent  the  appearance 
of  arithmetical  works  that  were  quite  as  worthless  as  any  of  earlier 
years.  There  appeared  others,  on  the  other  hand,  which  possessed  no 
little  merit  and  became  very  popular.  As  examples  of  the  latter  we 
would  mention  the  arithmetics  of  the  two  brothers,  Benjamin  D.  and 
Frederick  J^merson,  both  of  whom  were  well-known  teachers  in  Boston. 

The  arithmetics  of  later  days  are  combinations  of  the  old,  as  found  in 
our  early  arithmetics,  and  the  new  as  found  in  the  works  of  Colburn, 
For  example :  Our  old  arithmetics  generally  rejected  reasoning,  but 
gave  rules ;  Colburn's  books  reject  rules,  but  encourage  reasoning.  The 
better  class  of  our  later  arithmetics  contain  rules,  but,  at  the  same  time, 
give  demonstrations  and  encourage  students  to  think. 

About  the  year  1825  or  1830,  the  French  notation  of  numbers  began 
rapidly  to  displace  the  English.  Large  numbers  came  to  be  marked  off 
in  periods  of  three  digits  instead  of  six.  The  earliest  book  in  which  we 
have  noticed  the  adoption  of  the  French  notation  is  Robert  Patterson's 
edition  of  Dilworth's  School-master's  Assistant,  Philadelphia,  1805;  the 
latest  in  which  we  have  seen  the  English  notation  used  is  M.  Gibson's 
revised  edition  of  Abijah  and  Josiah  Fowler's  Youth's  Assistant,  Jones- 
borough,  Tenn.,  1850.  Some  of  our  recent  books  explain  both,  but  use 
the  French. 

It  is  a  rather  curious  fact  that  the  process  of  cancellation  did  not  come 
to  be  generally  used  in  our  arithmetics  before  about  1850.  In  1840  C. 
Tracy  published  aa  arithmetic  in  which  cancelling  was  freely  used,  a 
feature  which  was  then  "  entirely  peculiar  to  this  treatise,"  and  which 
distinguished  it  "from  all  others."  John  L.  Talbott's  Practical  Arith- 
metic (Cincinnati,  1853)  gives  the  "cancelling  system,"  but  only  in  the 
appendix,  and  remarks  in  the  preface  to  it,  "  In  Europe  this  system 
has  been  very  generally  adopted  in  the  higher  schools,  and  in  this  coun- 
try it  is  fast  becoming  known — and,  so  far  as  it  is  known,  it  supersedes 
the  usual  modes  of  operation,"  Charles  Davies  takes  pains  to  remark 
on  the  title-page  of  his  University  Arithmetic  (1857)  that  "  the  most 
improved  methods  of  analysis  and  cancellation^^  have  been  employed. 

The  order  in  which  the  various  arithmetical  subjects  havQcometobe 
taught  has  been  generally  improved  upon.  Federal  money  and  com- 
pound interest  no  longer  precede  common  and  decimal  fractions,  but 
come  after  them.  Fractions  have  been  moved  much  further  to  ward  the 
front  part  of  our  books.  The  placing  of  fractions  toward  the  end  of 
arithmetics  had  been  due  to  the  fact  that  the  majority  ofpupils  in  olden 
times  did  not  pursue  mathematics  long  enough  to  master  fractions,  and 
were  thus  put  through  a  course  in  arithmetic  with  only  integral  num- 
bers. Those  who  did  study  fractions  were  made  to  learn  the  rules  of 
interest  and  proportion  over  again  "  in  vulgar  fractions,"  and  then  again 
"  in  decimal  fractions."  Some  of  the  old  topics,  such  as  single  and 
double  position,  have  since  been  quite  generally  dropped,  but  we  think- 
that  there  is  still  room  for  improvement  in  that  respect.    Nothing  would 


INFLUX  OP  FEENCH  MATHEMATICS.  109 

be  lost  and  much  gained  if  alligation,  square  and  cube  root,  mensura- 
tion, and  some  of  the  more  difficult  applications  of  percentage  should 
be  dropped  from  our  arithmetics.  At  least  one  new  subject  has  been 
quite  generallyandjwethink,  appropriately  introduced  into  our  books — 
the  metric  system. 

Before  the  time  of  Colburn,  mental  arithmetic  was  quite  unknown  in 
our  schools.  Since  then  mental  and  written  arithmetic  have  not  always 
been  so  closely  united  as  they  should  be.  The  methods  used  in  the  two 
were  frequently  quite  diverse.  Too  often  they  were  taught  almost  like 
distinct  sciences,  so  that  a  pupil  might  bo  quite  proficient  in  the  one 
without  knowing  anything  of  the  other. 

Grube's  method  of  teaching  numbers  to  children  has  been  in  use 
among  us,  especially  in  the  East,  but  has  never  been  generally  adopted. 
It  is  such  a  refined  method  that  few  teachers  possess  the  skill  to  apply 
it  readily.  The  method  has  a  desirable  tendency  to  train  ready  and 
rapid  calculators,  and  has  much  to  commend  itself  to  teachers.* 

Since  the  beginning  of  this  century  arithmetic  has  come  to  be  re- 
garded as  the  most  important,  because  the  most  practical,  science  in 
our  elementary  schools.  Every  farmer  wished  his  sons  to  be  good  cal- 
culators ;  every  business  man  desired  to  be  "quick  at  figures;"  hence 
its  value  was  high  in  the  estimation  of  all.  Bookmakers  were  quick  to 
profit  by  this  sentiment.  They  began  to  multiply  the  number  of  test- 
books  in  the  course  until  there  were  two  books  in  mental  arithmetic, 
and  three  in  written,  in  several  of  the  series  in  general  use.  As  a  rule, 
the  examples  in  our  arithmetics  have  not  been  well  graded ;  difficult 
examples  have  been  introduced  so  early  in  the  course  as  to  embarrass 
and  discourage  even  the  best  students.  Many  examples  were  regular 
puzzles,  not  only  to  young  boys  and  girls,  but  to  almost  any  one  not 
trained  in  algebra.  There  are  numerous  problems  that  should  never 
have  found  a  place  in  our  arithmetics.  Wo  could  quote  from  arithme- 
tics dozens  a-nd  dozens  of  such  problems,  but  we  shall  give  only  one. 

The  137th  problem  of  the  miscellaneous  questions  in  the  third  part 
of  Emerson's  North  American  Arithmetic,  published  in  1835,  is  as  fol- 
lows: 

If  12  oxen  eat  up  3|  acres  of  grass  in  4  weeks,  and  21  oseu  eat  up  10  acres  in  9 
■weeks,  how  many  oxen  will  eat  up  24  acres  in  18  weeks;  the  grass  being  at  first 
equal  on  every  acre,  and  growing  uniformly. 

The  idea  of  placing  a  problem  of  such  difficulty  in  a  book  for  boys 
and  girls!  The  history  of  this  problem  in  this  country  shows  very 
plainly  that  it  is  beyond  the  power,  not  only  of  pupils,  but  even  of 
teachers  of  arithmetic.  Many  teachers  whose  minds  had  been  trained 
by  the  study  of  algebra  and  geometry  and,  perhaps,  even  higher 
branches  of  mathematics,  wrestled  with  it  in  vain.  There  existed  so 
much  uncertainty  regarding  its  true  solution  that  a  i^remium  of  fifty 

"  For  further  information  on  Grube's  method,  see  Prof.  T.  H.  Safford's  monograpl) 
on  Mathematical  Teaching,  pp.  19. 


110  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

dollars  was  offered  in  June,  1835,  for  the  most  "  lucid  analytical  solu- 
tion "  of  this  question.  A  committee  was  appointed,  with  I^.  Mackin- 
tosh as  chairman,  to  examine  the  solutions  presented  and  award  the 
prize.  The  committee  reported  112  solutions  received,  of  which  only 
48  gave  the  true  answer,  and  awarded  the  prize  to  Mr.  James  Robin- 
son, principal  of  the  department  of  arithmetic,  Bowdoin  School,  Bos- 
ton.* 

Think  of  it!  Out  of  112  of,  presumably,  the  best  arithmeticians  in 
the  country,  ouly  48  got  correct  results ;  and  yet  this  problem  was  in- 
tended to  be  solved  by  boys  and  girls. 

But  the  history  of  our  problem  is  not  yet  complete.  Nearly  twenty- 
five  years  later  a  revision  of  Mr.  Eobiuson's  solution  was  submitted  to 
the  National  Teachers'  Association,  at  Washington,  by  the  Hon.  Finley 
Bigger,  then  Register  of  the  CJ.  S.  Treasury;  it  was  referred  to  the  Math- 
ematical Monthly  for  publication,  and  was  printed  in  Vol.  II,  No.  3, 
December,  1859,  pp.  82-85.  Mr.  Bigger  assumed,  "for  the  purpose  of 
elucidation,"  that  the  question  was  susceptible  of  two  constructions, 
and  obtained  two  answers  in  addition  to  the  true  one.  The  editor  of 
the  Monthly  appended  an  algebraic  solution,  and  showed  that  there 
was  only  one  answer  that  would  satisfy  all  the  conditions  of  the  prob- 
lem, and  that  Mr.  Bigger  was  wrong  in  his  conclusions. 

There  is  no  ambiguity  in  the  problem.  Twenty -three  years  later,  Dr. 
Artemas  Martin  published  several  solutions  of  the  problem  in  the  Mathe- 
matical Magazine.  Dr.  Martin  does  not  consider  Mr.  Robinson's  solu- 
tion very  "  lucid,"  and  pronounces  it  liable  to  at  least  one  other  objec- 
tion— it  makes  "  mincemeat"  of  the  oxen,  inasmuch  as  fractions  of  oxen 
occur  throughout  the  analysis  of  the  question. 

There  is  another  curious  fact  connected  with  the  history  of  this  prob- 
lem. Neither  Mr.  Emerson,  nor  the  committee,  nor  Mr.  Robinson,  nor 
Mr.  Bigger,  nor  the  National  Teachers'  Association,  nor  the  Mathe- 
matical Monthly,  alludes  to  the  fact  that  the  question  is  taken  from  the 
AritJimetica  Universalis  of  Sir  Isaac  Newton,  published  in  1704,  which 
contains  a  "  lucid  analytical  solution."  Mr.  Emerson's  statement  of  the 
problem  differs  from  that  of  Newton  in  this,  that,  owing  to  a  misprint, 
the  fraction  J  instead  of  ^  is  given  by  the  former  in  the  number  of  acres 
contained  in  the  first  pasture,  which  mistake  produces  the  absurd  result 
of  37II5  oxen,  instead  of  3G.  The  above  question  goes  by  the  name  of 
the  "  pasturage  problem." 

There  exists  a  general  feeling  among  mathematicians  and  educators 
that  the  teaching  of  arithmetic  hag  been  overdone  in  our  schools. 
Parents  have  desired  their  older  boys  to  be  good  mathematicians.  But 
they  failed  to  perceive  the  truth  that  the  best  review  of  arithmetic  con- 
sists in  the  study  of  algebra ;  they  looked  upon  algebra  as  utterly  des- 
titute of  value.    In  consequence  the  boys  have  been  made  to  waste 

•Hendricks's  Analyst,  Vol.  Ill,  p.  75;  also  the  Mathematical  Magazine,  edited  by 
Dr.  Artemas  Martin,  Vol.  I,  pp.  17  and  43. 


INFLUX  OF  FRENCH  MATHEMATICS.  Ill 

their  time  at  the  study  of  circulating  decimals,  diJEfieult  problems  in 
stocks  and  exchange,  in  general  average,  in  alligation  medial  and  alliga- 
tion alternate,  in  square  and  cube  root,  and  in  combinations  and  per- 
mutations. From  the  manner  in  which  these  subjects  have  been  treated 
in  our  arithmetics,  a  student  derives  very  little  mental  training  from 
them.  The  presentation  of  duodecimals  is  not  only  unphilosophical, 
but  decidedly  absurd. 

Protests  have  been  made  from  time  to  time  against  the  over-study  of 
arithmetic.  Thus  in  1866  the  Superintendent  of  Public  Instruction  of 
California  said  in  his  Eeport  (p.  119):  "The  crack  classes  are  the 
arithmetic  classes,  and  the  merits  of  a  whole  school  not  unfrequently 
rise  or  fall  with  the  exploits  of  the  first  class  in  arithmetic  on  '■  examina- 
tion day.'  Arithmetic  is  well  enough  in  its  place,  but  the  sky  is  not  a 
blackboard,  nor  are  mountains  all  made  of  chalk.  Children  have  facul- 
ties other  than  that  of  calculation,  and  they  need  to  be  exercised  on 
appropriate  subjects."  This  doubtless  voices  the  sentiments  of  many 
thinking  teachers.  Five  years  ago  the  writer  heard  Prof.  Simon  New- 
comb,  in  a  lecture  at  the  Johns  Hopkins  University,  protest  against  ex- 
isting practices  in  the  teaching  of  arithmetic. 

Says  Prof.  T.  H.  Safford,  of  Williams  College :  "  The  mathematics 
have  their  (disciplinary)  value,  and  a  very  high  one  it  is ;  but  the  lower 
mathematics,  especially  arithmetic,  have  been  overdone  in  a  certain 
direction ;  I  mean  that  of  riddles,  puzzles,  brain-spinning,  as  the  Ger- 
mans call  it.  While  our  boys  and  girls  are  given  problems  to  solve 
which  quite  exceed  their  thinking  powers— I  don't  suppose  I  could  ever 
have  gone  successfully  through  Greenleaf  s  l!^"ational  Arithmetic  till  I 
had  graduated  from  college — their  minds  are  quite  undeveloped  in  the 
power  of  observation,  and  they  are  often  imperfectly  trained  in  the  four 
ground  rules,  especially  in  decimal  fractions."* 

A  very  remarkable  and  encouraging  step  toward  reform  was  taken 
in  1887  by  the  Boston  School  Board.  It  passed  the  following  orders  con- 
cerning the  study  of  arithmetic  :t 

"  1.  Home  lessons  in  arithmetic  should  be  given  out  only  in  excep- 
tional cases. 

"  2.  The  mensuration  of  the  trapezoid  and  of  the  trapezium,  of  the 
prism,  pyramid,  cone,  and  sphere ;  compound  interest,  cube  root  and 
its  applications;  equation  of  payments,  exchange,  similar  surfaces,  met- 
ric system,  compound  proportion,  and  compound  partnership,  should 
not  be  included  in  the  required  course. 

"  3.  All  exercises  in  fractions,  commission,  discount,  and  proportion, 
should  be  confined  to  small  numbers,  and  to  simple  subjects  and  pro- 
cesses, the  main  purpose  throughout  being  to  secure  thoroughness,  ac- 
curacy, and  a  reasonable  degree  of  facility  in  plain  ordinary  ciphering. 

*  The  Developmeut  of  Astronomy  in  the  United  States,  1888,  p.  27, 
\The  Academy,  January,  1888,  article:  "Arithmetic  in  Boston  Schools,"  by  General 
Francis  A.  Walker,  President  of  the  Massachusetts  Institute  of  Technology. 


112  TEACHING   AND   HISTORY    OF    MATHEMATICS. 

"4.  In  '  practical  problems,'  and  in  examples  illustrative  of  arithmet- 
ical principles,  all  exercises  are  to  be  ayoided  in  which  a  fairly  intelli- 
gent and  attentive  child  of  the  age  concerned  would  find  any  consider- 
able difficulty  in  making  the  statement  which  is  preliminary  to  the 
performance  of  the  properly  arithm.etical  operations.  When  arithmet- 
ical work  is  put  into  the  form  of  i;)ractical  or  illustrative  problems,  it 
must  be  for  the  purpose  of  interesting  and  aiding  the  child  in  the  per- 
formance of  the  arithmetical  operations,  and  with  a  view  to  their  com- 
mon utility. 

'<  5,  In  oral  arithmetic  no  racing  should  be  permitted ;  but  the  dicta- 
tion should  be  of  moderate  rapidity. 

"6.  The  average  time  devoted  to  arithmetic  throughout  the  primary 
and  grammar  school  course  should  be  three  and  a  half  hours  a  week; 
and  in  the  third  primary  grade  not  more  than  two  hours,  and  in  the 
first  and  second  primary  grades  not  more  than  three  and  a  half  hours 
each  per  week." 

The  considerations  which  led  the  School  Board  to  introduce  these 
changes  are  admirably  set  forth  by  General  Francis  A.  Walker.  The  reg- 
ulation regarding  home  lessons  in  arithmetic  may  be  a  good  one  under 
the  conditions  existing  in  Boston  at  the  time  of  its  adoption,  but  can 
hardly  be  recommended  for  general  adoption.  It  sounds  somewhat 
arbitrary.  The  reasons  which  led  to  its  adoption  are,  (1)  a  tendency 
among  grammar  school  teachers  to  unduly  magnify  the  importance  of 
arithmetic;  (2)  the  injustice  done  as  between  pupil  and  pupil  by  giving 
home  lessons,  since  the  facilities  for  study  at  home  are  so  very  different 
among  pupils ;  (3)  the  absence  of  the  teacher  prevents  any  authorita- 
tive interposition  to  put  a  stop  to  excessive,  and  therefore  damaging, 
study  over  problems  in  the  lesson.  "In  the  old  flogging  days  of  the 
Army  and  Navy,"  says  General  Walker,  "  it  was  always  required  that 
the  surgeon  should  stand  hj,  to  feel  the  pulse  of  the  poor  wretch  under 
the  lash,  to  watch  the  signs  of  approaching  nervous  collapse,  and,  in 
his  discretion,  to  forbid  the  punishment  to  proceed  further.  But  in  the 
case  of  our  young  children  who  are  assigned  home  lessons  in  arithmetic, 
no  such  humane  provision  exists.  Were  the  work  being  done  in  the 
open  school  room,  the  severest  master  would,  when  he  saw  that  the 
child  did  not  understand  the  problem,  could  not  do  the  work,  and  that 
it  was  only  becoming  more  excited  and  fatigued  by  repeated  attempts, 
interpose  either  to  give  assistance  or  to  put  a  stop  to  the  exercise.  In  the 
case  of  home  lessons,  however,  the  ambitious  and  sensitive  child  finds 
no  relief.  The  work  may  go  on  long  after  the  child  should  have  been 
in  bed  until  a  state  is  reached  where  further  persistence  is  not  only  in 
the  highest  degree  injurious,  but  has  no  longer  any  possible  relation  to 
success." 

"  Eegarding  the  remaining  five  orders,  considered  as  a  body,"  says 
General  Walker,  "it  may  be  said  that  the  committee,  in  framing  them. 


INFLUX  OF  FEENCH  MATHEMATICS.  113 

were  actuated  by  the  belief  that  both  loss  of  time  and  misdirection  of 
effort,  with  even  some  positively  injurious  consequences,  were  involved 
in  the  teaching  of  arithmetic,  as  carried  on  in  some  of  the  Boston  schools. 
And  here  let  me  say,  to  prevent  misapprehension,  that  the  committee 
at  no  time  intended  to  reflect  on  the  schools  of  our  own  city  as  compared 
with  those  of  neighboring  cities  and  towns.  Personally,  I  believe  that 
the  teaching  of  arithmetic  has  been  more  humane  and  rational  of  late 
years  in  the  schools  of  Boston  than  in  those  of  most  New  England 
towns  and  cities.    What,  then,  are  the  faults  complained  of? 

"  First — That  the  amount  of  time  devoted  to  this  study  is  in  excess 
of  what  can  fairly  be  allotted  to  it,  in  the  face  of  the  demands  of  other 
and  equally  important  branches  of  study. 

''^■Secondly — That  the  study  of  arithmetic  is  very  largely  pursued  by 
methods  supposed  to  conduce  to  general  mental  training,  which,  in  a 
great  degree,  sacrifice  that  facility  and  accuracy  in  numerical  comiju- 
tations  so  essential  in  the  after-life  of  the  pupil,  whether  as  a  student 
in  the  higher  schools  or  as  a  bread-winner. 

"  Thirdly — That,  as  arithmetic  is  taught  in  many,  perhaps  in  most 
schools,  the  possible  advantages  of  this  branch  of  study,  even  as  a 
means  of  general  mental  training  and  of  the  development  of  the  reason- 
ing powers,  are,  whether  by  fault  of  the  text-book  or  of  the  individual 
teacher  or  of  the  standards  adopted  for  examination,  largely  sacrificed 
through  making  the  exercises  of  undue  dilficulty  and  complexity,  which 
not  only  destroys  their  disciplinary  value  but  becomes  a  means  of  posi- 
tive  injury." 

The  whole  paper  of  General  Walker  is  well  worth  reading.  In  on© 
respect,  however,  we  can  not  endorse  the  action  of  the  Board.  It  seems 
to  us  that  the  metric  system  should  be  retained,  even  if  the  tables  of 
apothecaries'  weights  and  fluid  measure,  and  of  the  mariner's  measure, 
had  to  be  omitted  to  make  room  for  it.  The  memorizing  of- the  tables 
in  the  metric  system  is  not  difficult.  Moreover,  what  problems  offer 
better  opportunities  for  a  good,  thorough  course  in  the  use  of  decimal 
fractions  than  those  involving  meters  and  decimeters. 

But  there  is  still  another  reason  for  urging  the  spread  of  a  knowledge 
of  the  metric  system  in  elementary  schools.  If  the  masses  have  once 
acquired  sufficient  knowledge  and  familiarity  with  it  as  to  see  its  trans- 
cending superiority  over  the  old  traditional  tables  of  weights  and  meas- 
ures now  in  use,  then  we  may  look  forward  more  hopefully  to  the  early 
approach  of  the  time  when  the  French  weights  and  measures  will  be 
declared  the  only  legal  ones  in  the  United  States. 

European  nations  that  are  generally  regarded  as  being  much  more 
conservative  than  our  own,  have  introduced  them,  to  the  exclusion  of 
older  ones.  Even  the  miniature  republic  of  Switzerland  has,  within  the 
last  ten  years,  adopted  the  metric  system.  The  change  was  brought 
about  without  serious  inconvenience. 
881— No.  3 8 


114  TEACHING   AND    HISTOEY    OF   MATHEMATICS. 

UNITED   STATES  MILITAEY  ACADEMY.* 

In  1817  began  a  new  epoch  in  the'history  of  the  United  States  Mili- 
tary Academy.  At  this  time  Maj.  Sylvanns  Thayer  became  superin- 
tendent, and  under  him  the  Academy  entered  upon  a  career  of  unusual 
prosperity.  Thayer  was  a  native  of  Massachusetts,  graduated  at  Dart- 
mouth College,  and  then  entered  the  Military  Academy  as  a  cadet  in 
1807o  He  was  appointed  lieutenant  in  the  corps  of  engineers  in  1808. 
At  the  close  of  the  War  of  1812  he  was  sent  abroad  by  the  Government 
to  look  into  the  military  systems  of  Europe,  particularly  of  France. 
After  his  return  the  Academy  was  reorganized  according  to  French 
ideals,  but  without  discarding  entirely  English  teachings.  Prof.  Charles 
Davies  says  that  in  the  construction  of  the  course  of  study  at  West 
Point,  "  the  beautiful  theories  of  the  French  were  happily  combined 
with  the  practical  methods  of  the  English  systems,  and  the  same  has 
since  been  done,  essentially,  in  the  schools  of  England  and  France.'? 
Maj.  B.  C.  Boynton,  in  his  History  of  West  Point,  summarizes  the 
services  of  Major  Thayer  in  the  following  manner :  "  The  division  of 
classes  into  sections,  the  transfers  between  the  latter,  the  weekly  ren- 
dering of  class  reports,  showing  the  daily  progress,  the  system  and 
scale  of  daily  marks,  the  establishment  of  relative  class  rank  among  the 
members,  the  publication  of  the  Annual  Eegister,  the  introduction  of 
the  Board  of  Visitors,  the  check-book  system,  the  prepondering  influ- 
ence of  the  blackboard,  and  the  essential  parts  of  the  regulations  for 
the  Military  Academy  as  they  stand  to  this  day,  are  some  of  the  evi- 
dence of  the  indefatigable  efforts  of  Major  Thayer  to  insure  method,  or- 
der, and  prosperity  to  the  institution.  It  was  through  the  agency  of 
Major  Thayer  that  Prof.  Claude  Crozet,  the  parent  of  descriptive  geome- 
try in  America,  and  one  of  the  first  successful  instructors  in  higher 
mathematics,  permanent  fortifications,  and  topographical  curves,  be- 
came attached  to  the  Academy."  Crozet  had  been  a  French  officer  un- 
der ISTapoleon,  and  a  pupil  at  the  Polytechnic  School  in  Paris. 

Thayer  was  superintendent  at  West  Point  from  1817  to  1833.  The 
great  reputation  which  the  Academy  obtained  was  chiefly  due  to  his 
efforts.  His  discipline  was  very  strict.  The  last  years  of  his  administra- 
tion were  years  of  trial  to  him.  It  is  said  that  his  discipline  was  counted 
too  stern,  and  that  he  was  not  sustained,  as  he  should  have  been,  at  the 
War  Department.  Difficulties  arose  between  him  and  the  President  of 
the  United  States,  resulting  in  hisleavingthe  Academy.  General  Francis 
H.  Smith  says  of  him :  f  "  Colonel  Thayer  held  the  reins  with  a  firm 
hand  during  his  entire  administration,  and  if,  at  times,  he  transcended 
the  limits  of  legitiiiate  authority,  no  private  pique  or  personal  interest 
swayed  his  judgment.    He  was  animated  by  the  single  desire  to  give 

•For  Official  Eegisters  of  the  Military  Academy  and  for  valuable  iiiformatiou  re- 
gardinp;  it,  we  are  indebted  to  the  kindness  of  V/.  C.  Brown,  First  Lieutenant  First 
Cavalry,  Adjutant. 

t  West  Point  Fifty  Years  Ago,  New  York,  1879,  p.  6, 


INFLUX  OP  FRENCH  MATHEMATICS.  115 

efSciency  to  his  discipline,  and  to  train  every  graduate  upon  the  high- 
est model  of  the  true  soldier." 

Andrew  EUicott  was  professor  of  mathematics  from  1813  to  1820. 
The  following  deseriptiou  of  him  applies  to  the  time  preceding  the 
arrival  of  Thayer.  Says  E.  D;  Mansfield :  "  There  are  some  who  will 
recollect  Professor  Ellicott  sitting  at  his  desk  at  the  end  of  a  long 
room,  in  the  second  story  of  what  was  called  the  Mess  Hall,  teaching 
geometry  and  algebra,  looking  and  acting  precisely  like  the  old-fash- 
ioned school-master,  of  whom  it  was  written, 

" '  And  still  they  gazed,  and  still  the  wonder  grew 
That  one  small  head  could  carry  all  he  knew.' 

"  In  the  other  end  of  the  room,  or  in  the  next  room,  was  his  acting 
assistant,  Stephen  H.  Long.  *  *  *  The  text-book  used  was  Hut- 
ton's  Mathematics,  and  at  that  time  the  best  to  be  had.  *  *  *  It 
was  a  good  text-book  then,  for  there  were  no  cadets  trained  to  pursue 
deeper  or  more  analytical  works." 

As  already  stated.  Superintendent  Thayer  caused  the  classes  to  be 
divided  into  sections.  Prom  the  reminiscences  given  by  John  H.  B. 
Latrobe,  who  entered  the  Academy  as  a  cadet  in  1818,  we  see  that  the 
various  sections  received  their  mathematical  instruction  from  assist- 
ants, and  that  the  professor  of  mathematics  occasionally  visited  the 
sections.  Mr.  Latrobe  says :  *  "  I  do  not  remember  upon  what  princi- 
ple our  class  of  one  hundred  and  seventeen  members  was  divided  into 
four  sections  -,  I  recollect,  however,  that  I  was  put  into  the  first  section. 
*  *  *  Our  recitation  room  was  next  the  guard  room,  on  the  first 
floor  of  the  North  Barracks.  Here,  on  a  rostrum,  between  two  win- 
dows, sat  Assistant  Professor  S.  Stanhope  Smith,  and  here,  with  the  first 
volume  of  Hutton's  Mathematics  in  hand,  I  began  my  West  Point  edu- 
cation.   *    *    * 

"  I  am  not  sure  that  we  had  desks,  but  rather  think  that  we  were 
seated  on  benches  against  the  wall,  with  a  blackboard  to  supply  the 
place  of  pen  and  ink  and  slates,  although  I  am  not  certain  about  the 
slates.  Generally  we  had  the  section  room  to  ourselves.  Sometimes, 
however,  Mr.  Ellicott  would  pay  us  a  visit  and  ask  a  few  questions, 
ending  with  giving  us  a  sum  in  algebra,  to  explain  what  was  meant  by 
'an  infinite  series,'  which  was  the  name  he  went  by  in  the  corps." 

"I  have,"  continues  Latrobe  (p.  29),  "  no  other  recollection  of  him  as 
an  instructor,  except  once  when,  while  learning  surveying,  we  were 
chaining  a  line  from  a  point  in  front  of  his  house  to  an  angle  of  Fort 
Clinton,  and  back  again.  Our  accuracy  quite  astonished  the  good  old 
professor,  to  whom  we  did  not  admit  that  it  was  owing  to  our  having 
used  the  same  holes  that  the  pins  had  made  in  going  and  returning. " 

Professor  Ellicott  died  at  West  Point  and  was  buried  in  the  cemetery 
there.     "  My  last  visit  to  it  as  a  cadet,  "  says  Latrobe,  "  was  when  I 


'Report  Association  of  Graduates  of  the  U.  S.  Military  Academy,  1887,  p.  8, 


116  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

was  ou  the  escort  that  fired  the  vollies  over  the  grave  of  Andrew  Elli- 
cott,  the  professor  of  mathematics  who  lies  buried  there.  " 

Of  Hutton's  Mathematics  Latrohe  says:  "  I  have  of  ten  heard  those 
who  have  been  more  recently  educated  at  West  Point  speak  dispar- 
agingly of  the  Huttonian  day,  as  though  any  one  could  have  graduated 
then."  That  this  was  not  the  case  becomes  evident  when  he  says,  ^'  that 
the  first  sifting  in  June,  1819,  of  my  one  hundred  and  seventeen  comrades 
Qf  the  year  before,  reduced  the  number  to  fifty-nine,  the  next  sifting  to 
forty-eight,  and  the  number  that  got  through  the  meshes  ofthe  sieve  was 
but  forty.  Ofthe  others,  some  resigned,  some  were '  turnetl  back '  to  go 
over  the  year's  course  a  second  time,  and  some  were  found  to  be  defi- 
cient altogether.  These  last  were  called,  in  the  parlance  ofthe  cadets, 
'  Uncle  Sam's  bad  bargains.' " 

Jared  Mansfield,  the  professor  of  natural  and  experimental  philosophy, 
outlived  Ellicott  by  ten  years.  Both  were  veteran  surveyors  and  math- 
ematicians. Mansfield  retired  from  his  chair  in  1828.  Mr.  Latrobe  says 
that  Colonel  Mansfield,  "  although  a  most  competent  instructor,  was 
very  near-sighted,  and  I  am  not  prepared  to  say  that  this  defect  was 
not  sometimes  taken  advantage  of. "  Professor  Church  (class  of  1828) 
says  of  him:  "Professor  Mansfield  at  my  time  was  very  old,  yet  quite 
enthusiastic  in  his  branch  of  study,  generally  a  mere  listener  to  demon- 
strations, complimentary  to  a  good  one,  but  coldly  silent  to  a  bad  one." 

The  great  impulse  to  the  study  Of  mathematics  at  West  Point  was, 
however,  due  to  younger  men.  One  of  these  was  Claude  Orozet.  After 
graduating  at  the  Polytechnic  School  in  Paris,  he  had  been  artillery 
officer  under  Napoleon.  From  1816  to  1817  he  was  assistant  professor 
of  engineering  at  the  Academy,  and  from  1817  to  1823  full  professor. 
E.  D.  Mansfield  has  given  us  some  iateresting  recollections  of  Crozet's 
earliest  teaching  at  West  Point.  The  Junior  class  of  1817-18  was  the 
first  class  which  commenced  thoroughly  the  severe  and  complete  course 
of  studies  at  the  Academy.  Of  Professor  Crozet,  Mansfield  says  that 
he  was  to  teach  engineering,  but  when  he  met  the  class  he  found  that 
he  would  have  to  teach  mathematics  first,  as  u^ot  one  of  them  had  had 
the  necessary  preliminary  training  iu  x)ure  mathematics  for  a  course  in 
engineering.  "The  surprise  of  the  French  engineer,  instructed  in  the 
Polytechnique,  may  well  be  imagined  when  he  commenced  giving  his 
class  certain  problems  and  instructions  which  not  one  of  them  could 
comprehend  and  perform." 

Among  the  preliminary  studies  we  find  that  descriptive  geometry 
was  included.  "We  doubt,"  says  E.  D.  Mansfield,  "whether  at  that 
time  more  thafi  a  dozen  or  two  professors  of  science  in  this  country 
knew  there  was  such  a  thing  j  certainly  they  never  taught  it,  and 
equally  certain  there  was  no  text-book  in  the  English  language." 
This  science,  founded  by  Monge,  was  then  scarcely  thirty  years  old. 
Crozet  meant  to  begin  by  teaching  this  branch,  but  a  new  difiicnlfcy 
arose.    Just  then  he  had  no  text-book  on  the  subject,  and  geometry 


INFLUX   OP   FKENCH   MATHEMATICS.  IIT 

could  not  be  taught  orally.    What  was  to  be  done  ?    ^'  It  was  here  at 
this  precise  time  that  Crozet,  by  aid  of  the  carpenter  and  painter,  in- 
troduced the  blackboard  and  chalk.    To  him,  as  far  as  we  know,  is  due  / 
the  introduction  of  this  simple  machine.    He  found  it  in  the  Polytech- 
nique  of  France."    (E.  D.  Mansfield).  . 

Crozet  was,  however,  not  the  first  one  to  use  the  blackboard  in  this 
country.  Of  Eev.  Samuel  J.  May,  of  Boston,  it  is  said  that,  ''  to  the 
work  of  teaching  a  ijublic  school  he  then  brought  one  acquisition  which 
was  novel  in  that  day,  and  which  it  lias  taken  a  half  century  to  intro- 
duce into  elementary  schools,  private  and  public—a  knowledge  of  the 
uses  of  the  blackboard,  which  he  had  seen  for  the  first  time  in  1813  in 
the  mathematical  school  hept  by  Rev.  Francis  Xavier  Brosius,  a  Catholic 
priest  of  France,  who  had  one  suspended  on  the  wall  with  lumps  of 
chalk  on  a  ledge  below  and  cloth  hanging  on  either  side.'^  *  One  thing 
is  certain :  The  blackboard  was  introduced  in  this  country  by  French- 
men. Its  importance  in  the  school  room  can  hardly  be  overestimated. 
Simple  and  inexpensive  as  it  is,  its  introduction  into  our  colleges  was 
not  instantaneous.  For  geometrical  teaching  large  tablets  with  printed 
diagrams  were  used  in  our  best  colleges  long  after  Crozet  had  taught 
its  use  at  West  Point. 

Crozet,  says  E.  D.  Mansfield,  did  not  more  than  half  understand 
English.  "With  extreme  difiBculty  he  makes  himself  understood  and 
with  extreme  difficulty  his  class  comprehend  that  two  planes  at  right 
angles  with  one  another  are  to  be  understood* on  the  same  surface  of 
the  blackboard,  on  which  are  represented  two  different  projections  of 
the  same  subject."  The  first  problems  were  drawn  and  demonstrated 
on  the  blackboard  by  the  professor ;  afterward  they  were  drawn  and 
demonstrated  by  the  pupils,  and  then  carefull}"  copied  into  accurate 
drawings. 

In  1821  Crozet  published  his  Treatise  on  Descriptive  Geometry,  for 
the  use  of  cadets  of  the  U.  S.  Military  Academy  (New  Tork).  The 
first  87  pages  were  given  to  the  elementary  principles,  and  the  next  63 
pages  to  the  application  of  descriptive  geometry  to  spherics  and  conic 
sections.  This  is,  according  to  our  information,  the  first  English  work 
of  any  importance  on  descriptive  geometry,  and  the  first  work  pub- 
lished in  this  country  which  exhibits  to  the  student  that  gem  of  geom- 
etry— Pascal's  Theorem. 

Crozet  has  been  called  the  father  of  descriptive  geometry  in  this 
country.  He  taught  this  as  preparatory  to  engineering.  It  may  justly 
be  said,  also,  that  the  course  of  military  science  was  greatly  developed 
by  him. 

Mr.  Latrobe  favors  us  with  the  following  recollections  of  him :  "  There 
are  persons  whose  appearance  is  never  effaced  from  the  memory.  Of 
this  class  was  the  professor  of  the  art  of  engineering.  Col.  Claude 
Crozet,  a  tall,  somewhat  heavily-built  man,  of  dark  complexion,  black 

*  "American  Educational  Biography,"  Barnard's  Journal,  Vol.  XVI,  p.  141,  1866.    , 


118  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

hair  and  eyebrows,  deep-set  eyes,  remarkable  for  tlieir  keen  and  bright 
espression,  a  firm  mouth  and  square  chin,  a  rapid  speech  and  strong 
French  accent.  I  can,  even  after  the  lapse  of  between  sixty  and  seventy 
years,  fancy  that  I  see  the  man  before  me.  He  had  been  an  engineer 
under  Napoleon  at  the  battle  of  ^  Wagram  and  elsewhere,  and  the  anec- 
dotes with  which  he  illustrated  his  teaching  were  far  more  interesting 
than  the  'Science  of  War  and  Fortification,'  which  was  the  name  of  our 
text  book  at  the  time.  When  he  left  the  Academy  he  became  chief 
engineer  of  the  State  of  Virginia,  which  is  indebted  to  him  for  the  sys- 
tem that  made  her  mountain  roads  the  best,  tben,  in  America.  Per- 
haps my  recollection  of  Colonel  Crozet  is  strengthened  by  my  having 
seen  him  long  after  I  ceased  to  be  his  pupil." 

Ellicott  was  succeeded  in  the  professorship  of  mathematics  by  David 
B.  Douglass.  He  held  it  till  1823,  when  he  was  transferred  to  the  de- 
partment of  engineering,  where  he  taught  till  1831.  Professor  Church 
(class  of  1828)  says  of  him  :  "  Professor  Douglass,  of  engineering,  had 
the  reputation  of  being  an  able  engineer  and  a  fine  scholar,  yet  he 
was  by  no  means  a  clear  demonstrator.  His  style  was  diffuse  and  there 
was  a  great  want  of  logical  sequence  In  his  language.  Most  of  the 
course  of  engineering  was  given  to  the  class  by  him  from  the  black- 
board." He  was  afterward  the  chief  engineer  of  the  Croton  water- 
works. 

One  of  the  text-books  mentioned  by  E.  D.  Mansfield  as  having  been 
used  was  the  Mechanitjs  of  Dr.  Gregory  {•'  Old  Greg."),  who  was  pro- 
fessor at  the  Eoyal  Military  Academy  at  Woolwich.  His  works  are 
collections  of  rules  rather  than  expositions  of  principles,  and  are  want- 
ing in  analysis.  Gregory  is  at  his  best  when  ho  descends  to  the  mi- 
nutiae of  practice.  For  several  years  no  adequate  text-book  was  found 
for  civil  engineering.  In  1823  Major  O'Conner  translated  a  Treatise  on 
the  Science  of  War  by  De  Vernon,  which  had  been  prepared  in  1805  by 
the  order  of  the  French  Government,  and  was  the  text-book  in  the  poly- 
technic school.  This  translation  was  used  at  the  Academy  for  several 
years.  "  It  was  a  miserable  translation,"  says  General  Francis  H.  Smith, 
"but  it  was  the  best  that  could  be  had,  and  each  member  of  the  first 
class  was  required  to  take  a  copy,  costing  some  $20." 

After  being  vacated  by  Douglass,  the  chair  of  mathematics  was  taken 
by  one  whose  name  became  known  to  nearly  every  school- boy  in  our 
laud — Charles  Davies.  He  was  a  native  of  Connecticut,  graduated  at 
the  Academy  in  1815,  and  then  was  made  assistant  professor  of  mathe- 
matics. He  held  the  full  professorship  for  fourteen  years,  until  1837. 
He  earned  for  himself  a  wide  reputation,  not  as  an  original  investigator 
in  mathematics,  but  as  a  teacher  and  as  a  compiler  of  popular  text- 
books. He  was  always  described  by  his  pupils  as  an  excellent  instructor. 
Professor  Church  (class  of  1828)  says  of  him :  "Professor  Davies  was 
then  young,  enthusiastic,  a  clear  and  logical  demonstrator,  and  an  admir- 
able teacher.  He  had  at  once  imbibed  the  st)irit  and  fully  sympathized  in 


INFLUX  OF  FEENCH  MATHEMATICS.  119 

the  desires  of  the  superintendent,  and  labored  earnestly  to  carry  them 
out  in  building  up  a  logical  system  of  instruction  and  recitation,  which 
required  not  only  a  thorough  understanding  of  the  details  of  and  rea- 
sons for  everything  proposed,  but  a  clear,  concise,  and  complete  exam- 
ination of  it.'^ 

When  Church  was  a  cadet,  according  to  his  own  statement,  the  meth- 
ods of  instruction  were  entirely  new,  and  test-books  very  imperfect. 
The  professors  and  teachers  had  themselves  to  learn  the  true  use  of  the 
blackboard,  and  the  strict  and  detailed  manner  of  demonstration.  In 
algebra  the  best  text-book  that  could  be  obtained  was  a  poor  translation 
of  Lacroix,  In  geometry  we  had  a  translation  of  Legendre ;  in  trigo- 
nometry, a  translation  of  Lacroix ;  in  descriptive  geometry,  a  small  work 
by  Crozet,  containing  only  the  elements  without  application  to  the  inter- 
section of  surfaces  or  to  warped  surfaces.  These,  with  the  whole  of 
shades,  shadows,  and  perspective,  stone-cutting,  and  problems  in  engi- 
neering, were  given  by  lecture  to  the  class.  Emotes  were  taken  by  the 
cadets,  the  drawings  made  in  our  rooms  before  the  next  morning,  then 
presented  for  examination,  and  at  once  recited  upon  previous  to  the  fol- 
lowing lecture.  The  sections  in  mathematics,  philosophy,  and  engi- 
neering were  of  twenty  cadets  each,  and  were  kept  in  three  hours  daily. 
Biot's  work  on  analytical  geometry  was  used,  and  Lacroix's  calculus. 

Those  who  have  toiled  over  Davies'  text-books  may  enjoy  the  follow- 
ing reminiscences  of  him  :  "  Don't  you  remember,"  says  General  F.  H. 
Smith  (class  of  1833),  "when  muttering  out  an  imperfect  answer  to  one 
of  his  questions,  how  he  would  lean  forward  with  one  of  his  significant 
smiles  and  say,  '  How's  that,  Mr.  Bliss  ?'  But  I  will  not  now  dwell  upon 
his  long  and  faithful  career  in  the  department  of  mathematics.  The  re- 
sults of  his  labors  are  to  be  seen  in  the  distinguished  career  of  his  pupils 
and  in  his  series  of  mathematical  text-books,  which  are  as  household 
words  everywhere  in  the  United  States." 

When  John  H.  B.  Latrobe  was  a  cadet,  Davies  was  as  yet  only  assist- 
ant professor.  Latrobe  speaks  of  him  as  follows :  "  My  next  professor  of 
mathematics,  in  my  second  year's  course,  was  one  that  I  have  no  difficulty 
in  describing  and  whom  I  can  never  forget,  Charles  Davies.  Personally 
and  mentally  he  was  a  remarkable  man.  Of  the  middle  size,  with  a 
bright,  intelligent  face,  characterized  by  projecting  upper  teeth,  which 
procured  for  him  the  name  of  'Tush'  among  the  cadets,  his  whole  figure 
was  the  embodiment  of  nervous  energy  and  unyielding  will.  His  fear- 
less activity  at  a  fire  which  happened  in  a  room  in  the  South  Barracks, 
in  1819,  added  the  name  of  'Eush'  to  the  other.  He  was  a  kindly 
natured  man,  too,  and  the  patient  perseverance  that  he  devoted  to  the 
instruction  of  his  class  was  not  the  least  remarkable  feature  of  his 
character.  It  was  with  Professor  Davies  that  I  began  the  study  of 
descriptive  geometry,  for  which  no  books  in  English  had  then  been 
published.  He  had  no  assistance  beyond  the  blackboard  and  his  own 
intimate  knowledge  of  the  subject  and  faculty  of  oral  explanation.    For- 


120  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

tunately  this  was  exceptionally  great,  and  even  then  there  was  no  little 
amount  of  actual  labor  requisite  to  enable  the  pupil  to  understand  the 
difference  betweeen  the  horizontal  and  vertical  planes  and  the  uses  to 
be  made  of  them.  It  is  to  Professor  Davies  that  I  have  always  at- 
tributed in  a  great  measure  my  subsequent  successes  at  West  Point, 
and  hence  this  especial  notice  of  him  as  a  tribute  to  his  memory.  A 
much  more  enduring  tribute  is  that  awarded  by  the  countless  benefici- 
aries, the  colleges,  schools,  and  individuals  who  have  profited  by  his 
numerous  publications  in  connection  with  mathematical  science." 

Professor  Davies  taught  for  many  years  before  he  conceived  the  idea 
of  issuing  a  series  of  text-books.  Some  of  his  books — as  his  Legeudre 
and  Bourdon — were  adaptations  from  French  works,  modified  to  sup- 
ply the  wants  of  our  schools ;  others  were  prepared  on  his  own  plan. 
While  connected  with  the  Academy  as  professor  he  published  his 
Descriptive  Geometry,  1826  (a  more  extensive  work  than  Crozet's); 
Brewster's  Translation  of  Legendre,  1828  j  Shades,  Shadows,  and  Per- 
spective, 1832  5  Bourdon's  Algebra,  1834^  Analytical  Geometry,  1836^ 
Differential  and  Integral  Calculus,  1836 ;  a  Mental  and  Practical  Arith- 
metic. Overwork  in  the  preparation  of  these  textbooks  caused  bron- 
chial affection,  which  forced  him  to  resign  his  professorship  in  1837. 
He  visited  Europe  and  soon  after  his  return  occupied  the  professorship 
of  mathematics  at  Trinity  College,  Hartford,  Conn.,  but  ill-health  again 
induced  him  to  exchange  the  position  for  that  of  paymaster  in  the 
Army  and  treasurer  at  West  Point.  These  offices  he  resigned  in  1845. 
In  1848  he  became  professor  of  mathematics  and  philosophy  in  the 
University  of  New  York,  but  in  the  following  year  he  retired  to  Fishkill 
Landing,  on  the  Hudson,  that  he  might  have  leisure  to  complete  his 
series  of  text-books.  After  teaching  in  the  normal  school  at  Albany,  he 
was  made  professor  of  higher  mathematics  at  Columbia  College,  in  1857. 

In  1839  appeared  his  Elementary  Algebra;  in  1840,  his  Elementary 
Geometry  and  Trigonometry;  in  1846,  his  IJniversity  Arithmetic ;  in 
1850,  his  Logic  of  Mathematics ;  in  1852,  his  Practical  Mathematics ;  in 
1855,  jointly  with  William  G.  Peck,  a  Mathematical  Dictionary. 

Davies'  series  constituted  a  connected  mathematical  course,  from 
primary  arithmetic  up  to  calculus.  His  books  were,  as  a  rule,  perspic- 
uous, clear,  and  logically  arranged.  They  were  not  too  difficult  for  the 
ordinary  student,  and  contained  elements  of  great  popularity.  The 
original  editions  would  be  found  quite  inadequate  for  the  wants  of 
schools  of  the  present  day.  "The  first  translations  of  Bourdon  aud 
Legendre  were  imperfect "  (Prof.  C.  W.  Sears,  class  of  1837).  Davies 
himself  greatly  modified  some  of  his  text-books  in  later  editions.  In 
his  revisions  he  was  greatly  aided  by  his  son-in-law,  Prof.  William  G. 
Peck.  The  most  recent  revisions  are  those  made  by  Prof.  J.  Howard 
Van  Amringe,  of  Columbia  College. 

Brewster's  Legendre  underwent  some  changes  in  the  hands  of  Davies. 
In  the  original  work,  as  also  in  the  translation  of  Brewster  and  Farrar, 


INFLUX  OF  FEENCH  MATHEMATICS.  121 

each  proposition  was  enunciated  with  reference  to  and  by  aid  of  the  par- 
ticular diagram  used  for  the  demonstration.  But  Davies  gave  the 
propositions  without  reference  to  particular  figures  and,  to  that  extent, 
returned  to  the  method  of  Euclid.  In  later  editions  Davies  did  not  use 
Brewster's  translation,  but  took  the  original  and  translated  and  adapted 
it  to  the  courses  in  American  schools.  In  trigonometry  he  was  wedded 
to  the  line  system. 

The  reasoning  sometimes  employed  by  Professor  Davies  in  his  books 
has  been  found  to  be  open  to  objection.  This  is  certainly  true  of  his 
treatment  of  iufinite  series.  In  his  Legendre  the  treatment  of  the 
circle  is  not  such  as  will  carry  conviction  to  the  young  mind.  Thus,  he 
says  in  one  edition,  that  "  the  circle  is  but  a  regular  polygon  with  an 
infinite  number  of  sides."  *  A  trained  mathematician  who  feels  that  he 
can  give  more  rigorous  proofs  by  sounder  methods,  whenever  he  may 
wish  to  do  so,  will  employ  this  idea  of  the  circle,  and  of  curves  in  gen- 
eral, with  profit  and  satisfaction.  After  much  study  he  may  even 
arrive  at  the  conviction  that  the  method  of  limits  and  that  of  infini- 
tesimals are  essentially  alike.  But  it  is  the  experience  of  the  majority 
of  our  teachers  that  the  infinitesimal  method  and  the  treatment  of  the 
circle  as  a  polygon  appear  to  beginners  as  enigmatical  and  obscure. 
Of  our  more  recent  geometries,  the  best  and  the  most  popular  have 
abandoned  those  methods. 

Kor  is  Davies'  explanation  of  a  limit  and  of  the  first  differential 
co-eificient  satisfactory.  Listen  to  the  testimony  of  one  of  his  pupils  :t 
"  I  had  not  been  a  teacher  of  the  calculus  long  *  *  *  before  I  dis- 
covered that  I  had  almost  everything  to  learn  respecting  it  as  a  rational 
system  of  thought.  Difficulties  were  continually  suggested  in  the 
course  of  my  reflection  on  this  subject  about  which  I  had  been  taught 
nothing,  and  consequently  knew  nothing.  I  found,  in  short,  that  I  had 
only  been  taught  to  work  the  calculus  by  certain  rules,  without  know- 
ing the  real  reasons  or  principles  of  those  rules ;  pretty  much  as  an 
engineer,  who  knows  nothing  about  the  mechanism  or  principle  of  an 
engine,  is  shown  how  to  work  it  by  a  few  superficial  and  unexplained 
rules." 

It  is  our  opinion  that  under  Professors  Davies  and  Church  the  philos- 
ophy of  mathematics  was  neglected  at  West  Point.  If  this  criticism 
be  true  of  West  Point,  which  was  for  several  decennia  unquestionably 
the  most  influential  mathematical  school  in  the  United  States,  how  much 
more  must  it  be  true  of  the  thousands  of  institutions  throughout  the 
country  which  came  under  its  influence  ?  If  this  stricture  were  not  cor- 
rect, then  such  a  book  as  Bledsoe's  Philosophy  of  Mathematics  would 
never  have  been  written ;  there  would  have  been  no  occasion  for  it. 

Of  Davies'  assistants,  we  shall  mention  Lieutenant  Boss.  General  F. 
H.  Smith  says:  "  There  was  associated  with  Professor  Davies,    *    *    * 

*  Davies'  Legendre,  1856,  Book  V,  Scholium  to  Proposition  XII. 
tProf.  A.  T.  Bledsoe,  Philosopliy  of  Mathematics,  1867,  p.  214,  note. 


122  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

as  his  chief  assistant  in  mathematics,  having  charge  of  the  fourth  class, 
Lieut.  Edward  C.  Boss,  of  the  class  of  1821.  He  was  the  best  teacher 
of  mathematics  I  ever  knew,  and  it  is  singular,  too,  that  he  had  no  fac- 
ulty of  demonstration.  He  gave  to  our  class  many  extra  discussions  in 
the  difficult  points  in  algebra,  particularly  on  what  he*called  the  final 
equations,  for  he  was  not  pleased  with  Farrar's  translation  of  La  Croix, 
our  text-book  in  algebra,  and  he  was  preparing  his  translation  of  Bour- 
don. In  putting  upon  the  blackboards  these  extra  demonstrations 
every  line  appeared  as  if  it  had  been  printed,  so  neat  was  he  in  the  use 
of  his  chalk  pencil.  But  when  he  commenced  to  explain  he  would  twist 
and  wriggle  about  from  one  side  of  the  board  to  the  other,  pulling  his 
loDg  whiskers,  and  spitting  out,  in  inordinate  volumes,  his  tobacco  juice. 
The  class  was  as  ignorant  when  he  closed  as  when  he  began.  We  copied, 
word  for  word,  what  was  written,  well  knowing  that  on  the  next  day 
the  first  five  would  be  called  upon  to  make  the  discussion.  We  read  to 
him  what  we  had  placed  on  the  board.  Then  commenced  his  power  as 
a  teacher.  In  a  series  of  orderly  questions  he  would  bring  out  the 
points  of  the  discussion,  step  by  step,  sometimes  occupying  half  an  hour 
with  each  cadet,  and  when  the  three  hours  of  recitation  were  over  w© 
knew  the  subject  thoroughly.  He  was  an  expert  in  his  power  of  ques- 
tioning a  class.  He  did  this  without  note  or  book,  and  gave  such  earn- 
estness and  vividness  to  his  examinations  that  he  kept  his  class  up  to 
the  highest  pitch  of  interest  all  the  time." 

General  Smith  gives  us  a  description  also  of  Courtenay.  "  Edward 
H.  Courtenay,  who  graduated  at  the  head  of  Ross's  class,  was  our  pro- 
fessor of  natural  and  experimental  philosophy,  fifty  years  ago.  There 
never  was  a  clearer  minded — a  more  faithful  teacher — or  a  more  modest 
one  than  Professor  Courtenay.  Well  do  I  remember  the  hesitating  man- 
ner with  which  he  would  correct  the  grossest  error  on  the  part  of  a 
member  of  his  section — I  hardly  think  so.  He  resigned  his  professorship 
in  1834,  and  after  holding  many  offices  of  high  dignity,  as  professor  and 
civil  engineer,  he  was  elected  professor  of  mathematics  in  the  Univer- 
sity of  Virginia  in  1842." 

Courtenay  was  instructor  at  West  Point  from  1821  to  1834,  excepting 
the  four  years  from  1824  to  1828.  During  his  first  years  of  teaching  he 
was  assistant  professor  of  natural  and  experimental  philosophy,  and 
then  assistant  professor  of  engineering.  After  the  resignation  of  Jared 
Mansfield  he  was  appointed  professor  of  natural  and  experimental  phil- 
osophy, and  acted  in  that  capacity  for  five  years.  In  1833  he  trans- 
lated from  the  French  of  M.  Boucharlat  an  elementary  treatise  on  me- 
chanics, and  made  additions  and  emendations. 

The  chair  of  military  and  civil  engineering,  made  vacant  by  the  resig- 
nation of  Professor  Douglass,  was  filled  by  the  appointment  of  Lieut. 
Dennis  H.  Mahan.  Mahan  graduated  at  West  Point  in  1824,  holding 
the  first  place  in  his  class  of  thirty-one  members.  "After  remaining  at 
the  Academy  as  an  instructor  for  two  years,  he  was  ordered  to  Europe 


INFLUX  OF  FRENCH  MATHEMATICS.  123 

to  study  public  engineering  works  and  military  institutions.  By  special 
favor  of  the  French  Ministry  of  War,  Lieutenant  Mahan  was  allowed  to 
join  the  Military  School  of  Application  for  Engineers  and  Artillerists  at 
Metz,  where  he  remained  for  more  than  a  year,  under  the  instruction  of 
men  whose  names  were  then,  and  are  now,  widely  known  in  science."* 
When,  after  his  return,  he  entered  upon  the  duties  of  his  department  at 
West  Point,  he  supplemented  the  meagre  volume  of  O'Connor  with  ex- 
tensive notes.  These  notes  developed  into  his  well-known  treatises  on 
Civil  Engineering  and  Field  Fortifications. 

Such  is  the  brief  record  of  the  professional  career  of  Professor  Ma- 
han ;  liut  it  fails  to  convey  any  adequate  idea  of  the  influence  which 
he  exerted  upon  engineering  science  in  this  country.  To  ai^preciate 
this,  it  must  be  remembered  that  for  many  of  those  forty-one  years 
(during  which  he  was  professor)  West  Point  was  our  only  school  of 
mathematical  and  physical  science  where  the  rigid  requirements  and 
high  standard  now  deemed  essential  were  even  attempted.  Every  offi- 
cer  of  the  present  corps  of  engineers  who  has  served  long  enough  to 
win  reputation  in  the  performance  of  the  civil  duties  assigned  to  that 
corps,  and  many  of  the  eminent  civil  engineers  of  the  country  as  well, 
now  gratefully  remember  how,  before  those  old  blackboards  in  that  un- 
pretending recitation  room  at  West  Point,  they  learned  from  Professor 
Mahan,  with  the  rudiments  of  their  profession,  a. high-toned  discipline 
and  the  fundamental  truth  that  without  precision  of  ideas,  rigid  analy- 
sis, and  hard  work  there  can  be  no  such  thing  as  success. 

"But  if  civil  engineering  owes  much  to  our  late  colleague,  military 
engineering  and  the  science  of  war  owe  more.  For  many  years,  and  up 
to  the  day  of  his  death,  he  was  in  that  branch  of  the  profession  con- 
fessedly the  highest  scholastic  authority  in  America."* 

The  death  of  Mahan  was  pathetic.  In  his  last  years  he  often  had  fits 
of  melancholy,  and,  in  an  instant  of  acute  insanity,  he  plunged  from  a 
steamer  into  the  Hudson  and  drowned. 

Professor  Davies  was  succeeded  in  the  chair  of  mathematics  by  Prof. 
Albert  E.  Church.  Church  was  a  native  of  Connecticut,  graduated  at 
West  Point  in  1828,  served  as  assistant  professor  from  1828  to  1831, 
also  from  1833  to  1837.  He  was  then  acting  professor  of  mathematics 
for  about  a  year,  and  in  1838  he  became  full  professor,  retaining  the 
chair  till  his  death  m  1878.  He  published  four  works  which  have  been 
used  considerably  in  A  merican  colleges.  His  Differential  and  Integral 
Calculus,  1842,  was  more  extensive  than  that  of  Davies.  In  the  new 
edition  of  1851  a  chapter  on  the  Elements  of  the  Calculus  of  Variations 
was  inserted.  In  1851  appeared  his  Analytical  Geometry,  in  which  he 
followed  somewhat  the  work  of  Biot  on  this  subject.  In  1857  his  Plane 
and  Spherical  Trigonometry  was  published.  In  1865  appeared  his  El- 
ements of  Descriptive  Geometry,  in  the  preparation  of  which  he  was 

*  Biographical  Memoirs  of  the  National  Academy  of  Sciences,  Vol.  II,  1886,  p.  32. 


124  TEACHING   AND    HISTORY   OF   MATHEMATICS. 

aided  by  the  French  works  of  Leroy  and  Oliver,  and  by  the  elaborate 
American  work  of  Warren.  Later  editions  give  also  the  application  of 
the  subject  to  shades,  shadows,  and  perspective.  The  Descriptive 
Geometry  met  with  larger  sales  than  any  of  his  other  works. 

As  a  teacher,  Professor  Church  is  spoken  of  by  General  F.  H.  Smith  as 
follows :  "  Prof.  Albert  E.  Church  was  an  assistant  professor  of  math- 
ematics when  my  class  entered  in  1829.  He  occasionally  heard  my  sec- 
tion in  the  third  class  course  and  exhibited  then  the  clearness  and 
perspicuity  which  marked  his  long  career  as  a  professor  of  mathematics." 

Prof.  Arthur  S.  Hardy  (class  of  1869)  gives  the  following  remin- 
iscences of  the  mathematical  teaching  in  his  day :  * 

"  The  class  was  divided  into  sections  of  from  ten  to  tifceen.  The  alpha- 
betical arrangement,  first  adopted,  became  in  a  few  weeks  a  classifica- 
tion by  scholarship—transfers  up  and  down  being  made  weekly.  The 
descent  was  easy,  but  it  was  hard  to  rise  a  section.  The  last  section  we 
called  Leis  Immortels  (lazy  mortals?).  In  each  section  each  student 
recited  daily.  The  sections  were  taught  by  army  officers  detailed  at 
Professor  Church's  request.  The  latter  had  no  section,  but  generally 
visited  each  daily.  Each  recitation  was  one  hour  and  a  half  long. 
Professor  Church's  visits  were  dreaded.  He  usually  asked  questions. 
His  questioning  was  searching.  He  was  a  stickler  for  form — it  was 
not  enough  to  mean  right. 

"Personally  he  did  not  inspire  me^  he  had  no  magnetism — was  dry 
as  dust,  as  his  text-books  are.  He  delivered  one  lecture  on  the  calculus. 
I  never  got  a  glimpse  of  the  philosophy  of  mathematics— of  its  history, 
methods  of  growth.  The  calculus  was  a  machine,  where  results  were 
indisputable,  but  its  mechanism  a  mystery.  I  do  not  think  he  had  a 
great  mathematical  mind.  It  was  geometrical,  rather  than  an  analytic 
one.  A  problem,  which  he  and  Professor  Bartlett  once  attacked  to- 
gether, the  latter  solved  by  a  few  symbols  on  a  piece  of  paper,  while  the 
former  drew  a  diagram  with  his  cane  on  the  gravel— to  Bartlett's  dis- 
gust, who  despised  geometry.  Church's  text-books  are  French  adapta- 
tions, minus  the  luminousuess  and  finish  of  form  of  French  text-books. 

"  The  only  instance  of  Church's  being  disconcerted  was  on  being  told 
by  a  cadet  that  the  reason  for  -j-  becoming  —  in  passing  through  zero 
was  that  the  cross-piece  got  knocked  off  in  going  through.  You  can 
imagine  that  the  would-be  wit  was  placed  in  arrest. 

"  The  mathematical  recitation  at  West  Point  was  a  drill-room.  In  my 
judgment  its  result  was  a  soldier  who  knew  the  manoeuvres,  but  it  did 
not  give  an  independent,  self-reliant  grasp  of  methods  of  research.  In 
descriptive  geometry,  the  Academy  had  a  magnificent  collection  of 
models,  but  they  were  shown  us  after  the  study  was  finished — in  other 
words,  mental  discipline  was  the  object — practical  helps  and  ends  were 
secondary.    Great  changes  have  been  made  since." 

William  H.  C.  Bartlett  (class  of  1826)  was  assistant  professor  for 


'Letter  to  the  -writer,  November  12,  1888. 


I 


INFLUX  OF  FRENCH  MATHEMATICS.  125 

several  years  at  West  Point'.  He  was  permanently  appointed  professor 
of  natural  and  experimental  philosophy  in  1836.  In  1871  he  was  re- 
tired from  military  service  at  his  own  request,  and  shortly  after  he 
accepted  the  place  of  actuary  for  the  Mutual  Life  Insurance  Company 
of  E"ew  York. 

The  need  of  an  astronomical  observatory  being  felt  at  West  Point, 
Professor  Bartlett  went  abroad  in  1840  to  order  instruments  and  visit 
observatories.  On  his  return  it  was  necessary  to  provide  room  for  the 
instruments  in  the  new  library  building  of  the  school,  on  account  of 
the  great  prejudice  existing  in  Congress  against  a  separate  observatory.* 

Bartlett  published  treatises  on  Optics,  1839 ;  Acoustics ;  Synthetic 
Mechanics,  1850;  Analytical  Mechanics,  1853  5  Spherical  Astronomy, 
1855,  He  contributed  also  to  Silliraan's  Journal.  His  Analytical  Me- 
chanics is  the  first  American  work  of  its  kind  which  starts  out  with, 
and  evolves  everything  from,  that  precious  intellectual  acquisition  of 
the  nineteenth  century— the  laws  of  the  indestructibility  of  matter  and 
energy.  Dr.  E.  S.  McCulloch  (who,  by  the  way,  rewrote  Bartlett's 
Mechanics  without  allowing  his  own  name  to  appear  anywhere  in  the 
revised  edition)  says:t  "More  than  thirty  years  ago,  at  West  Point, 
Professor  Bartlett,  in.  his  treatise  on  Analytical  Mechanics,  still  used 
there  as  a  test-book,  had  deduced  the  whole  science  from  one  single 
equation,  or  formula,  well  known  to  every  cadet  as  his  equation  A  / 
and  he  thus  expressed  and  discussed  fully  what  now  is  generally  called 
the  Law  of  the  Conservation  of  Energy." 

Bartlett's  successor  was  Prof  Peter  S.  Michie,  the  present  incumbent 
in  the  chair.  Michie  graduated  in  1863,  and  has  been  instructor  there 
since  1867.  He  has  published  Wave  Motion,  Relating  to  Sound  and 
Light,  1882  ;  Hydrostatics ;  and  Analytical  Mechanics,  1886.  The  first 
edition  of  the  last  treatise  was  never  published ;  the  second  edition, 
1887,  differs  considerably  from  the  first.  It  is  on  the  plan  of  Bartlett's 
book  on  the  same  subject,  but  it  is  confined  to  mechanics  of  solids.  It 
contains  also  a  good  introduction  to  graphical  statics,  a  subject  which, 
in  recent  years,  has  come  to  be  studied  in  this  country.  The  first  to 
place  a  treatise  on  graphical  statics  in  the  hands  of  American  engi- 
gineers  was  A.  Jay  Du  Bois,|  professor  at  Lehigh  University,  Pa. 
This  subject  owes  its  development  chiefly  to  Culmann,  who,  in  1866 
published  in  Zurich  his  Graphische  Statik.  In  technical  schools  in 
Europe  this  method  has  been  favorably  received.  In  this  country 
original  contributions  of  great  value  have  been  made  to  this  subject 
by  Prof.  Henry  T.  Eddy,  of  the  University  of  Cincinnati.§ 

*  The  Development  of  Astronomy  intlie  United  States,  by  Prof.  T.  H.  Safford,  18S8, 
p.  19. 

t  Papers  read  before  tlie  New  Orleans  Academy  of  Sciences,  1886-87,  Vol.  I.,  No. 
1,  p.  120. 

t  The  Elements  of  Graphical  Statioa  and  their  Application  to  Framed  Structures, 
New  York,  1875. 

§  Van  Nostrand's  Engineering  Magazine,  1878.  Article:  "A  New  General  Method 
in  Graphical  Statics." 


126  TEACHING   AND    HISTORY    OP   MATHEMATICS. 

In  1841  Professor  Church  was  aided  iu  his  department  by  five  assist- 
ants. This  number  has  been  increased  since,  and  is  now  nine.  These 
assistants  have  been,  we  believe,  always  selected  from  young  graduates 
of  the  Academy.  The  course  of  study  in  pure  ahd  applied  mathematics 
was,  in  1841,  as  follows :  Fourth  Class  (first  year),  Davies'  Bourdon, 
Legendre,  and  Descriptive  Geometry;  Third  (7/as«,  Davies' works  on 
Shades  and  Shadows,  Spherical  Projections  and  Warped  Surfaces,  Sur- 
veying, Analytical  Geometry,  and  Calculus ;  Second  Class,  Courtenay's 
Boucharlat's  Traits  de  Mecanique,  Eoget's  Electricity,  Magnetism, 
Electro  Magnetism  and  Electro-Dynamics,  Bartlett's  Optics,  Gummere'a 
Astronomy;  First  Class,  Mahan's  Treatises  on  Field  Fortifications, 
Lithographic  Notes  on  Permanent  Fortification,  Attack  and  Defence, 
Mines  and  other  Accessories,  Composition  of  Armies,  Strategy,  Course 
of  Civil  Engineering,  Lithographic  Notes  on  Architecture,  Stone  Out- 
ting,  Mechanics  (studied  by  the  first  section  only). 

As  Church's  and  Bartlett's  text-books  came  from  the  press  they  were 
introduced  in  place  of  earlier  ones.  Thus,  Davies'  Geometry,  Calculus, 
Descriptive  Geometry,  and  Trigonometry,  Gummere's  Astronomy, 
and  Courtenay's  Boucharlat's  Mechanics  were  gradually  displaced  by 
new  books.  But  some  of  Davies'  books  have  been  retained  to  the 
present  day.  We  may  here  state  that  the  power  of  selecting  text-books 
does  not  lie  with  each  individual  professor,  but  with  the  Academic 
Board. 

After  the  death  of  Mahan,  in  1871,  the  chair  of  military  and  civil 
engineering  was  given  to  Junius  B.  Wheeler,  of  the  class  of  1855.  He 
retired  in  1884,  and  was  succeeded  by  James  Mercur.  Professor  Wheeler 
gradually  substituted  books  of  his  own  in  place  of  Mahan's  treatises. 

Professor  Church's  successor  is  Prof.  Edgar  W.  Bass,  of  the  class  of 
1868.  By  him  more  attention  is  given  to  the  philosophical  exposition 
of  fundamental  principles  than  was  given  by  his  predecessors.  Davies' 
and  Church's  textbooks  are  still  used,  but  they  are  much  modified  by 
copious  notes  by  Professor  Bass.  In  calculus  the  notation  of  Leibnitz 
has  always  been  used,  but  now  the  Modern  is  also  given.  At  present 
the  calculus  is  based  upon  the  Newtonian  conception  of  rates,  but  his 
notation  is  not  used.  In  1879  determinants  and  least  squares  were 
introduced  into  the  course  of  study.  Peck's  Determinants  and  Chau- 
venet's  Least  Squares  being  the  text-books  used. 

The  present  mode  of  instruction  in  mathematics  involves  recitations 
by  cadets  at  the  blackboard,  lectures  and  explanation  of  the  text,  numer- 
ous applications  of  each  principle,  and  written  recitations  by  the  stu- 
dents. The  sections  number  from  nine  to  twelve  students,  for  one  and 
a  half  hour's  instruction.  Three  hours  are  allotted  for  the  study  of  each 
mathematical  lesson.    Recitations  are  daily,  Sundays  excepted. 

The  course  of  study  for  1888  is  as  follows  :  Fourth  class,  Davies'  ele- 
ments of  Algebra,  Legendre's  Geometry,  Ludlow's  Elements  of  Trigo- 
liometry,  Davies'  Surveying,  Church's  Analytical  Geometry  j  Third  class, 


INFLUX  OF  FRENCH  MATHEMATICS.  127 

Ohnrch's  Analytical  Geometry,  Descriptive  Geometry  with  its  applica- 
tions to  Spherical  Projections,  Bass's  Introduction  to  ths  Differential  Cal- 
culus, Church's  Calculus,  Church's  Shades,  Shadows,  and  PerspectivCj 
Chauvenet's  Treatise  on  the  method  of  Least  Squares;  Second  class, 
Michie's  Mecbanics,  Bartlett's  Astronomy,  Michie's  Elements  of  Wave 
Motion  relating  to  Sound  and  Light;  First  class,  Wheeler's  Civil  Engi- 
neering, Field  Fortifications,  Mercur's  Mahan's  Permanent  Fortifications 
(edition  of  1887) ;  Wheeler's  Military  Engineering  (Siege  Operations  and 
Military  Mining),  Elements  of  the  Art  and  Science  of  War,  and  Mahan's 
Stereotomy.  For  reference,  is  used  a  book  called  Eoyal  Engineers' 
Aide-M6moire,  Parts  I  and  IL 

It  may  be  stated,  in  conclusion,  that  the  U.  S.  Military  Academy  has 
contributed  to  the  educational  force  of  the  country  no  less  than  thirty- 
five  presidents  of  universities  or  colleges,  twenty-seven  principals  of 
academies  and  schools,  eleven  regents  and  chancellors  of  educational 
institutions  and  one  hundred  and  nineteen  professors  and  teachers,  mak- 
ing a  total  of  one  hundred  and  ninety-two  instructors  of  youth  distrib- 
uted throughout  the  country.* 

HARVAKD  COLLEGE. 

In  1807  John  Farrar  succeeded  Samuel  Webber  in  the  chair  of 
mathematics  and  natural  philosophy.  Farrar  was  a  native  of  Massa- 
chusetts. After  graduatingat  Harvard  he  studied  theology  at  Andover, 
but  having  been  appointed  tutor  of  Greek,  in  1805,  he  never  entered  the 
ministerial  office.  He  retained  his  chair  till  1836,  when  he  resigned  in 
consequence  of  a  painful  illness  that  finally  caused  his  death.  He  was 
a  most  amiable,  social,  and  excellent  man,  and  endeared  to  his  friends. 
By  the  students  he  was  familiarly  called  "  Jack  Farrar." 

Prof.  Andrew  P.  Peabody  gives  the  following  reminiscences  of  him :  f 
"  He  delivered,  when  I  was  in  college,  a  lecture  every  week  to  the  Junior 
class  on  natural  philosophy,  and  one  to  the  Senior  class  on  astronomy. 
His  were  the  only  exercises  at  which  there  was  no  need  of  a  roll-call. 
Ko  student  was  willingly  absent.  The  professor  had  no  notes,  and 
commenced  his  lecture  in  a  conversational  tone  and  manner,  very  much 
as  if  he  were  explaining  his  subject  to  a  single  learner.  But  whatever 
the  subject,  he  very  soon  rose  from  prosaic  details  to  general  laws  and 
principles,  which  he  seemed  ever  to  approach  with  blended  enthusiasm 
and  reverence,  as  if  he  were  investigating  and  expounding  divine  mys- 
teries. His  face  glowed  with  inspiration  of  his  theme.  His  voice,  which 
was  unmanageable  as  he  grew  warm,  broke  into  a  shrill  falsetto;  and 
with  the  first  high  treble  notes  the  class  began  to  listen  with  breathless 
stillness,  so  that  a  pin-fall  could,  I  doubt  not,  have  been  heard  through 
the  room.    This  high  key  once  reached  there  was  no  return  to  the  lower 


•Annual  Report  of  the  Board  of  Visitors  to  the  U.  S.   Military  Academy  made  to 
the  Secretary  of  War,  for  the  year  1886. 
tHarvard  Reminlscencea,  by  Andrew  P.  Peabody,  Boston,  1838,  p.  70, 


128  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

notes,  nor  any  intermission  in  the  outflow  and  quickening  rush  of  lofty 
thought  and  profound  feeling,  till  the  bell  announced  the  close  of  the 
hour,  and  he  piled  up  all  the  meaning  that  he  could  stow  into  a  parting 
sentence,  which  was  at  once  the  climax  of  the  lecture,  and  the  climax 
of  an  ascending  scale  of  vocal  utterance  higher,  I  think,  than  is  within 
the  range  of  an  ordinary  soprano  singer.  I  still  remember  portions  of 
his  lectures,  and  they  now  seem  to  me  no  less  impressive  than  they  did 
in  my  boyhood."* 

Josiah  Quincy  t  says  in  hi&  diary,  which  he  kept  while  a  student  at 
college,  that  by  the  prolixity  of  Professor  Everett  in  his  lectures,  "  we 
gained  a  miss  from  Farrar  for  the  fourth  time  this  term.  This  was 
much  to  the  gratification  of  the  class,  who  in  general  hate  his  branch, 
though  they  like  him." 

Professor  Farrar  did  not  distinguish  himself  by  original  research  in 
mathematics,  but  he  was  prominent  and  among  the  first  to  introduce 
important  reforms  in  the  mathematical  teaching  in  American  colleges. 
He  was  the  first  American  to  abandon  English  authors  and  to  place 
translations  of  Continental  works  on  mathematics  in  the  hands  of  stu- 
dents in  the  New  World. 

In  1818  appeared  Farrar's  Introduction  to  the  Elements  of  Algebra, 
selected  from  the  Algebra  of  Euler.  Notwithstanding  the  transcend- 
ing genius  of  Euler  as  a  mathematician  and  the  high  estimation  he  was 
held  on  the  Continent,  his  algebra  was  scarcely  to  be  met  with  previous 
to  this  time,  either  in  America  or  England.  It  was  written  by  the  au- 
thor after  he  became  blind,  and  was  dictated  to  a  young  man  entirely 
without  education,  who  by  this  means  became  an  expert  algebraist.l 
Farrar's  Euler  was  a  very  elementary  book,  and  was  intended  for  stu- 
dents preparing  to  enter  college.  It  differed  from  the  English  works  in 
this,  that  it  taught  iDupils  to  reason,  instead  of  to  memorize  without  un- 
derstanding. 

In  the  same  year  appeared  also  Farrar's  translation  of  the  Algebra  of 
Lacroix,  which  was  first  published  in  France  about  twenty  years  pre- 
viously. Lacroix  was  one  of  the  most  celebrated  and  successful  teach- 
ers and  writers  of  mathematical  text-books  in  France.  Farrar  trans- 
lated also  Lacroix's  Arithmetic,  but  this  does  not  appear  to  have  been 

5 

*  Professor  Peabody  continues  his  reminiscences  as  follows :"  I  recall  distinctly  a. 
lecture  in  which  he  exhibited,  in  its  various  aspects,  the  idea  that  in  mathematical 
science,  and  in  it  alone,  man  sees  things  precisely  as  God  sees  them,  handles  the  very 
scale  and  compasses  with  which  the  Creator  planned  and  built  the  universe ;  another 
in  which  he  represented  the  law  of  gravitation  as  coincident  with,  and  demonstrative 
of,  the  divine  omnipresence  ;  another,  in  which  he  made  us  almost  hear  the  music  of 
the  spheres,  as  he  described  the  grand  procession,  in  infinite  space  and  in  immeasur- 
able orbits,  of  our  own  system  and  the  (so  called)  fixed  stars.  His  lectures  were 
poems,  and  hardly  poems  in  prose;  for  his  language  was  unconsciously  rythmical, 
and  his  utterances  were  like  a  temple  chant." 

t  Figures  of  the  Past,  p.  23. 

t  North  American  Eoview,  1818. 


INFLUX  OF  FRENCH  MATHEMATICS.  129 

received  as  favorably  iu  this  country  as  the  other  works  of  the  Harvard 
professor. 

In  1819  was  published  at  Cambridge -the  Geometry  of  that  famous 
French  mathematician,  Legendre.  A  similar  translation  was  made  in 
England  by  David  Brewster.  Legendre  has  been  the  greatest  modern 
rival  of  Euclid.  In  France,  in  most  schools  in  America,  and  in  some 
English  institutions,  the  venerable  and  hoary-headed  Euclid  was  made 
to  withdraw  and  make  room  for  Legendre. 

If  the  question  be  asked,  what  is  the  difference  between  the  geome- 
tries of  Euclid  and  Legendre,  we  would  answer  that  the  main  object  of 
Legendre  was  to  make  geometry  easier  and  more  palatable  to  students. 
This  he  succeeded  in  doing,  but  at  a  sacrifice  of  scientific  rigor.  The  fol- 
lowing are  the  principal  points  of  difference  between  Euclid  and  Legen- 
dre: (1)  Legendre  treats  the  theory  of  parallels  differently;  (2)  Legen- 
dre does  not  give  anything  on  proportion,  but  refers  the  student  to 
algebra  or  arithmetic.  The  objection  to  this  procedure  is  that  in  arith- 
metic and  algebra,  the  properties  of  proportion  are  unfolded  with  regard 
to  numbers,  but  not  with  regard  to  magnitude  in  general.  From  a 
scientific  point  of  view  this  is  a  serious  objection,  especially  if  we  remem- 
ber that  in  geometry  incommensurable  quantities  arise  quite  as  frequently 
as  commensurable  quantities  do.  Euclid's  treatment  of  proportion  dis- 
plays wonderful  skill  and  rigor,  but  is  very  difficult  and  abstract  for 
students  beginning  the  study  of  geometry;  (3)  Euclid  never  supposes 
a  line  to  be  drawn  until  he  has  first  demonstrated  the  possibility  and 
shown  the  manner  of  drawing  it.  Legendre  is  not  so  scrupulous,  but 
makes  use  of  what  are  called  "  hypothetical  constructions."  (4)  Legen- 
dre introduces  new  matter,  especially  in  solid  geometry,  changes  the 
order  of  propositions,  and  gives  new  definitions  (as,  for  instance,  his 
definition  of  a  straight  line). 

In  1820  Farrar  published  his  translation  of  Lacroix's  Trigonometry. 
The  original  gave  the  centesimal  division  of  the  circle,  but  in  the  trans- 
lation the  sexagesimal  notation  was  introduced.  This  trigonometry 
adopted  the ''line  system."  Bound  together  with  this  book  was  the 
"Application  of  Algebra  to  Geometry."  This  was  selected  from 
the  Algebra  of  Bezout.  Kegarding  this  selection  Professor  Farrar 
says :  "  It  was  the  intention  of  the  compiler  to  have  made  use  of  the 
more  improved  treatise  of  Lacroix  or  that  of  Biot  upon  this  subject; 
but  as  analytical  geometry  has  hitherto  made  no  part  of  the  mathemat- 
ics taught  in  the  public  seminaries  of  the  United  States,  and  as  only 
a  small  portion  of  time  is  allotted  to  such  studies,  and  this  is  in  many 
instances  at  an  age  not  sufficiently  mature  for  inquiries  of  an  abstract ' 
nature,  it  was  thought  best  to  make  the  experiment  with  a  treatise 
distinguished  for  its  simplicity  and  plainness."  * 

The  next  book  in  the  "  Cambridge  Course  of  Mathematics,"  as  Far- 
rar's  works  were  called,  was  an  Elementary  Treatise  on  the  Application. 

*  See  advertisement  to  the  Treatise. 
8Sl~No.  3 9 


130  TEACHING  AND   HISTOEY   OF  MATHEMATICS. 

of  Trigonometry  (1828),  in  the  preparation  of  which  were  used  Cag- 
noli's  and  Bonnycastle's  Trigonometries,  Delambre's  Astronomy,  Be- 
zout's  Navigation,  and  Puissant  and  Malortie's  Topography. 

In  1824  were  published  the  First  Principles  of  the  Differential  and  In- 
tegral Calculus,  "  taken  chiefly  from  the  mathematics  of  Bezout."  This 
is  the  first  text-book  published  in  America  on  the  calculus  and  employ- 
ing the  notation  of  Leibnitz.  It  is  based  on  the  infinitesimal  method. 
Bezout  flourished  in  France  before  the  Kevoiutiou.  His  works  were, 
therefore,  at  this  time,  rather  old,  but  his  calculus  was  selected  in  pref- 
erence to  others  "  on  account  of  the  plain  and  perspicuous  manner  for 
which  the  author  is  so  well  known,  as  also  on  account  of  its  brevity  and 
adaptation  in  other  respects  to  the  wants  of  those  who  have  but  little 
time  to  devote  to  such  studies."  * 

The  introduction  is  taken  from  Oarnot's  Reflexions^  and  gives  the  ex- 
planation by  the  "  compensation  of  errors." 

The  translation  of  Bezout's  calculus  is  only  in  part  the  work  of  Profes- 
sor Farrar.  After  having  begun  it,  he  was  obliged  to  go  to  the  Azores, 
on  account  of  the  health  of  his  wife,  and  the  translation  was  completed 
by  George  B.  Emerson.  He  had  it  printed  with  his  introduction  and 
notes,  so  that  when  Professor  Farrar  returned  he  found  it  ready  for  use 
in  the  college,  t 

Farrar's  translations  and  selections  from  French  authors  were  at  once 
adopted  as  text-books  in  some  of  our  best  institutions.  Several  books 
in  the  series  were  used  at  the  U.  S.  Military  Academy  and  at  the  Uni- 
versity of  Virginia. 

The  professor  of  mathematics  and  natural  philosophy  was  always  as- 
sisted by  tutors.  They  generally  taught  the  pure  mathematics  to  the 
lower  classes.  In  1825  there  were  three.  One  of  them,  James  Hay- 
ward,  had  been  tutor  for  five  years,  and  had  striven  to  reform  thelbeach- 
ing  of  elementary  geometry.  He  was  made  professor  in  1826,  but  a 
year  later  he  severed  his  connection  with  the  college  and  engaged  in 
civil  engineering,  in  which  he  became  a  high  authority.  The  original 
survey  of  the  Boston  and  Providence  Railway  was  made  by  him 
Among  the  other  tutors  of  note  who  served  during  the  time  of  Profes 
sor  Farrar  were  Thomas  Sherwin,  A.  P.  Peabody,  and  Benjamin  Peirce 
Dr.  Peabody  is  now  Plummer  professor  of  Christian  morals,  emeritus 
Among  Harvard  men  of  Farrar's  time  are  also  Charles  Henry  Davis 
who  afterward  served  on  the  Coast  Survey  and  established  the  Amer 
lean  Ephomeris,  and  Sears  Cook  Walker,  who,  later,  became  a  noted 
astronomer. 

We  now  proceed  to  inquire  into  the  terms  for  admission  and  the  courses 
of  study.  Since  1816  the  whole  arithmetic  has  been  required  for  ad- 
mission to  Harvard  College.  In  1819  a  trifling  amount  of  algebra  was 
added.    The  catalogue  of  1825  specifies  the  requirements  as  follows  : 

•Advertisement  to  the  translation. 

f  Barnaard's  Journal,  1878,  "  Schools  as  They  Should  Be,"  by  George  B.  Emerson. 


INFLUX  OF  FRENCH  MATHEMATICS.  131 

"Fundamental  rules  of  arithmetic;  vulgar  and  decimal  fractions ;  pro- 
portion, simple  and  compound;  single  and  double  fellowship;  alliga- 
tion, medial  and  alternate;  and  algebra  to  the  end  of  simple  equa. 
tions,  comprehending  also  the  doctrine  of  roots  and  powers,  and  arith- 
metical and  geometrical  progression."  The  books  used  in  the  exami- 
nation were  the  Cambridge  editions  of  Lacroix's  Arithmetic  and  Euler's 
Algebra.  In  1841  Euler's  Algebra  or  the  First  Lessons  in  Algebra  were 
required.  No  other  changes  were  made  until  1843.  The  catalogue  for 
that  year  mentions  for  admission  Davies'  First  Lessons  in  Algebra  to 
"  Extraction  of  Square  Root ;  "  and  "  An  Introduction  to  Geometry  from 
the  most  approved  Prussian  text-books,  to  VII. — Of  Proportions."  I^o 
other  subjects  were  added  until  1866-67,  though  there  were  some 
changes  in  the  text- books.  In  1850  Davies'  and  Hill's  Arithmetics  are 
mentioned ;  in  1853  Davies'  and  Chase's  Arithmetics ;  in  1859  Davies', 
Chase's,  or  Eaton's  Arithmetics,  Euler's  Algebra,  or  Davies'  First  Les- 
sons, or  Sherwin's  Common  School  Algebra,  and  the  Introduction  to 
Geometry ;  in  1865  Chase's  Arithmetic,  Sherwin's  Algebra,  Hill's  Sec- 
ond Book  in  Geometry,  Parts  I  and  II,  or  "  An  Introduction  to  Geom- 
etry as  the  Science  of  Form"  as  far  as  p.  130. 

In  addition  to  these  statements  taken  from  catalogues  it  will  be  in- 
teresting to  add  the  following  account,  given  by  Prof.  William  F.  Allen, 
of  the  class  of  1851 :  *  "  The  requirements  for  admission  were  not  much 
above  a  common  school.  That  is,  I  got  my  arithmetic  and  algebra  in  a 
country  district  school  (well  taught).  Geometry  I  picked  up  for  myself 
in  a  very  small  quantity.  I  remember  at  the  entrance  examination  I 
was  asked  what  an  angle  was.  I  thought  I  knew,  but  I  think  I  con- 
vinced the  examiner  that  I  didn't;  however,  I  got  in  clear." 

During  the  first  ten  or  eleven  years  of  his  teaching  Professor  Farrar 
used  the  books  of  Samuel  Webber.  A  second  edition  of  Webber's 
Mathematics  appeared  during  Professor  Farrar's  incumbency.  In  1818 
the  course  of  study  in  mathematics  was  as  follows :  t  FreshmeUj  alge- 
bra and  geometry,  during  the  first  and  second  term  and  three  weeks 
of  the  first  term.  iSophomores,  algebra,  trigonometry  and  its  appli- 
cations to  heights  and  distances,  and  navigation  during  the  third  term. 
Jujiiors,  natural  philosophy  and  astronomy  (Enfield's),  mensuration  of 
superficies  and  solids,  and  surveying  during  the  third  term.  In  place 
of  Hebrew,  on  the  written  request  of  their  parents  or  guardians,  stu- 
dents were  permitted  to  attend  to  mathematics  with  the  private  in- 
structor, or  Greek,  or  Latin,  or  French ;  Seniors,  conic  sections  and 
spherical  geometry  during  the  first  term  and  half  of  the  second.  We 
are  informed,  moreover,  that  for  the  attendance  on  the  private  in- 
stuctor  in  mathematics,  which  was  optional,  there  was  a  separate  charge, 
at  the  rate  of  $7.50  per  quarter. 

There  was  a  public  examination  of  each  class  in  the  third  term  and  a 

*  Letter  to  the  writer,  November  6,  1888.         f  North  American  Review,  March,  1818. 


132  TEACHING    AND    HISTORY    OP   MATHEMATICS. 

public  exhibition  of  performances  in  composition,  elocution,  and  in  the 
mathematical  sciences,  three  times  a  year.  Prizes  were  also  given. 
The  Bowdoin  prize  dissertation  was  read  in  the  Chapel  in  the  third 
term.  Of  these  prizes,  the  first  premium  was  given  in  1815  to  Jared 
Sparks,  of  the  Senior  class,  for  a  dissertation  "  On  the  character  of  Sir 
Isaac  Newton,  and  tlie  influences  and  importance  of  his  discoveries." 
'.rhe  title  of  this  essay  would  show  that  Sparks  had,  very  probably, 
studied  fluxions,  though  this  branch  was  not  included  in  the  cirricul urn 
for  1818,  given  above.  Fluxions  never  had  been  a  regular  study,  oblig- 
atory upon  all  the  students,  though  provisions  seem  to  have  been  made 
for  those  wishing  to  prosecute  it. 

During  the  twenty-nine  years  that  Farrar  was  professor,  from  1807  to 
1836,  275  theses  were  written  by  students  on  mathematical  subjects  at 
Harvard,  and  deposited  in  the  library  of  the  college.  Of  these  about 
one-fourth  contain  solutions  of  fluxional  problems  (or  problems  involving 
the  differential  and  integral  calculus) ;  a  little  less  than  one-half  are 
on  the  calculation  and  projection  of  eclipses ;  the  remaining  on  algebra, 
mechanics,  surveying,  etc.  Many  of  these  papers  are  interesting  me- 
morials of  men  since  become  in  different  degrees  famous.  Thus  George 
Bancroft  wrote,  in  1817,  a  thesis,  "  Invenire  Motum  Verum  Modorum 
Lunse  5  "  George  B.  Emerson,  on  "  Fluxional  Solutions  of  Problems  in 
Harmonicks  "  (1817) ;  Warren  Colburn,  on  "Calculation  of  the  Orbit  of 
the  Comet  of  1810  ; "  Sears  Cook  Walker,  in  1825,  on  "  The  Transit  of 
Venus  in  1882,"  and  "  The  Effect  of  Parallax  upon  the  Transit  in  1882; " 
Benjamin  Peirce,  in  1828,  on  "  Solutions  of  Questions  *  *  *  from 
the  Mathematical  Diary,  etc. ;  "  Wendell  Phillips,  in  1831,  on  "  Some 
Beautiful  Eesults  to  which  we  are  Led  by  the  Differential  Calculus  in 
the  Development  of  Functions."* 

The  catalogue  for  1820  shows  that  Webber's  Mathematics  and  Euclid's 
Geometry  had  been  discarded.  Farrar's  new  books  came  now  to  be 
used.  The  Freshmen  studied  Legendre's  Geometry  and  Lacroix's  Al- 
gebra. Analytic  methods  began  to  acquire  a  foothold.  Conic  sections 
were  displaced  by  analytic  geometry,  which,  with  trigonometry,  was 
begun  in  the  Sophomore  and  concluded  in  the  Junior  year.  The  Cata- 
logues from  1821  to  1824,  inclusive,  da  not  give  the  course  of  study.  In 
1824  the  Juniors  studied,  during  the  second  term,  differential  calculus 
from  Bezout's  work,  unless  they  exercised  their  j)rivilege  of  electing  mod- 
ern languages  in  place  of  mathematics.  The  catalogue  of  1830  shows 
some  slight  changes  in  the  course.  The  Freshmen  studied  Legendre's 
Plane  Geometry,  algebra,  solid  geometry ;  the  Sophomores,  trigonome- 
try and  its  applications,  topography,  and  caJciihis  ;  the  Juniors,  natural 
philosophy  and  mechanics  in  the  second  term,  and  electricity  and  mag- 
netism in  the  third  term ;  the  Seniors,  optics  and  natural  philosophy. 

The  following  remarks  by  Dr.  Peabody  applying  to  this  period  are 

*  "Mathematical  Theses  of  Junior  and  Senior  Classes,  1782-1839,  by  Henry  C.  Bad- 
ger," Bibliographical  Contributions  of  the  Library  of  Harvard  College,  No.  32. 


INFLUX  OF  FRENCH  MATHEMATICS.  133 

instructive :  "  The  chief  labor  and  the  crowning  honor  of  successful 
scholarship  were  in  mathematics  and  the  classics.  The  mathematical 
course  extended  through  the  entire  four  years,  embracing  the  differential 
calculus,  the  mathematical  treatment  of  all  departments  of  physical 
science  then  studied,  aod  a  thoroughly  mathematical  treatise  on  as- 
tronomy. (Gum  mere's,  afterward  replaced  by  Farrar's  almost  purely 
descriptive  treatise."*) 

The  year  1832  marks  an  epoch  in  the  history  of  mathematical  teach- 
ing at  Harvard.  It  was  then  that  Benjamin  Peirce  became  professor 
of  mathematics  and  natural  philosophy. 

While  there  had  been  men  in  this  country  who  had  cultivated  mathe- 
matics with  ardor,  they  had  seldom  possessed  the  talent  and  aspirations 
for  original  research  in  this  science.  We  have  had  many  who  were 
called  "  mathematicians,"  but  if  this  name  be  used  in  the  highest  sense, 
and  be  conferred  upon  only  such  persons  as  have  been  able  to  discover 
mathematical  truths  not  previously  known  to  man,  then  it  can  fall 
upon  very  few  Americans.  The  mere  ability  of  mastering  the  contents 
of  even  difficult  mathematical  books,  or  of  compiling  good  school-books 
in  this  science,  does  not  make  him  a  mathematician  worthy  of  standing 
by  the  side  of  Legendre,  the  Bernoullis,  Wallis,  Abel,  Tartaglia,  or 
Pythagoras — to  say  nothing  of  such  master  minds  as  Archimedes,  Leib- 
nitz, and  lifewton.  But  at  last  we  have  come  to  a  name  which  we  may 
pronounce  with  pride  as  being  that  of  an  American  mathematician. 
We  need  not  hesitate  to  rank  along  with  the  names  of  Wallis  and 
John  Bernoulli  that  of  Benjamin  Peirce. 

It  has  beeil  said  that  a  young  boy  detected  an  error  in  the  solution 
given  to  a  problem  by  Nathaniel  Bowditch.  "  Bring  me  the  boy  who 
corrects  my  mathematics,"  said  Bowditch.,  and  Benjamin  Peirce,  thirty 
years  later,  dedicated  one  of  his  great  works  "  To  the  cherished  and  re- 
vered memory  of  my  master  in  science,  Nathaniel  Bowditch,  the  father 
of  American  Geometry."  The  title  of  "father  of  American  Geometry," 
which  Peirce  confers  upon  his  beloved  master,  has  been  bestowed  by 
foreign  mathematicians  upon  Peirce  himself.  Sir  William  Thomson 
referred,  in  an  address  before  Section  A  of  the  British  Association,  to 
Peirce  as, "  the  founder  of  high  mathematics  in  America."  On  a  similar 
occasion  Arthur  Cayley  spoke  of  him  as  the  "father  of  American 
mathematics." 

Benjamin  Peirce  was  born  at  Salem  in  1809,  He  entered  Harvard 
College  at  the  age  of  sixteen,  and  devoted  himself  chiefly  to  mathemat- 
ics, carrying  the  study  far  beyond  the  limits  of  the  college  course. 
Thus  he  attended  lectures  on  higher  mathematics  by  Francis  Grund. 
While  an  under-graduate  he  was  a  pupil  of  Nathaniel  Bowditch,  who 
perceived  the  genius  of  the  young  man  and  predicted  his  future 
greatness,    Bowditch  directed  him  in  the  development  of  his  scientific 

*  Harvard  Eeminiscences,  by  A.  F.  Peabody,  p.  203. 


134  TEACHING  AND  HISTORY   OF  MATHEMATICS. 

powers,  and  gave  Mm  valaable  instruction  in  geometry  and  analysis. 
When  Bowditch  was  publishing  his  translation  and  commentary  of  the 
M^canique  Celeste,  Peirce  helped  in  reading  the  proof-sheets,  and  there- 
by contributed  greatly  toward  rendering  it  free  from  errors.  This  critical 
reading  of  that  great  work  of  Laplace  must  have  been  an  education 
to  him  in  itself.  Indeed,  a  great  part  of  Peirce's  scientific  labors  was 
in  the  field  of  analytic  mechanics. 

Dr.  Peabody  gives  the  following  reminiscences  of  Peirce  :*  "  While 
Benjamin  Peirce  the  younger  was  still  an  under-graduate  *  #  #  it 
was  said  that  in  the  class-room  he  not  infrequently  gave  demonstrations 
that  were  not  in  the  text-book,  but  were  more  direct,  summary,  or 
purely  scientific  than  those  in  the  lessons  of  the  day.  College  classes 
were  then  farther  apart  than  they  are  now;  but  even  in  our  Senior 
year  we  listened,  not  without  wonder,  to  the  reports  that  came  up  to 
our  elevated  platform  of  this  wonderful  Freshman,  who  was  going  to 
carry  off  the  highest  mathematical  honors  of  the  university.  On  grad- 
uating, he  went  to  Northampton  as  a  teacher  in  Mr.  Bancroft's  Eound 
Hill  School,  and  returned  to  Cambridge  in  1831  as  tutor.  The  next 
year  the  absence  of  Professor  Farrar  in  Europe  left  him  at  the  head  of 
the  mathematical  department  (which  he  retained  till  his  death),  the  fol- 
lowing year  receiving  the  appointment  of  professor ;  while  Mr.  Farrar 
on  his  return  was  still  unable  to  take  charge  of  class  instruction." 

In  1842  Peirce  was  appointed  Perkins  professor  of  astronomy  and 
mathematics.  This  position  he  held  until  his  death,  in  October,  1880. 
Tutor  Henry  Flint  is  the  only  person  ever  connected  with  the  college 
for  a  longer  time. 

We  shall  first  speak  of  the  mathematical  text  books  written  by  Peirce, 
then  of  his  record  as  a  teacher,  and,  lastly,  of  his  original  researches.! 

As  soon  as  he  entered  upon  his  career  as  teacher  of  mathematics  at 
Harvard  he  began  the  preparation  of  mathematical  text-books.  In 
183")  appeared  his  Elementary  Treatise  on  Plane  Trigonometry,  and  in 
1836  his  Elementary  Treatise  on  Spherical  Trigonometry.  The  two  were 
published  in  a  single  volume  in  later  editions.  In  1836  appeared  also 
his  Elementary  Treatise  on  Sound ;  in  1837,  his  Elementary  Treatise  on 
Plane  and  Solid  Geometry  and  his  Elementary  Treatise  on  Algebra  -, 
during  the  period  1841-46  he  wrote  and  published  in  two  volumes  his 
Elementary  Treati&e  on  Curves,  Functions,  and  Forces ;  in  1855,  was 
published  his  Analytical  Mechanics. 

Eev.  Thomas  Hill,  ex-President  of  Harvard  and  an  early  pupil  of 
Peirce,  speaks  of  these  books  as  follows :  "  They  were  so  full  of  novel- 
ties that  they  never  became  widely  popular,  except,  perhaps,  the  Trigo- 
nometry; but  they  had  a  permanent  influence  upon  mathematical  teach- 
ing in  this  country ;  most  of  their  novelties  have  now  become  common- 
places in  all  text-books.    The  introduction  of  infinitesimals  or  of  limits 

•Harvard  Reminiscences, p.  181. 

t  We  shall  draw  freely  from  the  Memorial  Collection,  by  Moses  King,  1881. 


INFLUX  OF  FRENCH  MATHEMATICS.  135 

into  elemeutary  books ',  the  recognition  of  direction  as  a  fundamental 
idea ;  the  use  of  Hassler's  definition  of  sine  as  an  arithmetical  quotient, 
free  from  entangling  alliance  with  the  size  of  the  triangle ;  the  similar 
deliverance  of  the  expression  of  derivative  functions  and  differential 
co-efficients  from  the  superfluous  introduction  of  infinitesimals;  the 
fearless  and  avowed  introduction  of  new  axioms,  when  confinement  to 
Euclid's  made  a  demonstration  long  and  tedious — in  one  or  two  of 
these  points  European  writers  moved  simultaneously  with  Peirce,  but 
in  all  he  was  an  independent  inventor,  and  nearly  all  are  now  generally 
adopted." 

The  ratio  system  in  trigonometry  was  used  before  this  by  Hassler  in 
his  masterly,  but  ill-appreciated,  work  on  Analytic  Trigonometry,  and 
also  by  Charles  Bonnycastle  in  his  Inductive  Geometry.  But  this  sys- 
tem met  with  no  favor  among  teachers.  The  most  popular  works  on 
trigonometry,  such  as  the  works  of  Davies  and  Loomis,  as  also  those 
of  Smyth,  Hackley,  Eobinson,  Brooks,  and  Olney,  adhered  to  the  old 
and  obsolete  "line  system,"  and  it  was  not  till  within  comparatively 
recent  years  that  the  "  ratio  system "  came  to  be  generally  adopted. 
The  old  "line  system"  was  brought  to  America  from  England,  but  the 
English  discarded  it  earlier  than  we  did.  In  1849  De  Morgan  wrote 
that  the  old  method  of  defining  trigonometric  terms  was  universal  in 
England  until  very  lately. 

The  final  victory  of  the  system  in  this  country  is  due  chiefly  to  the 
efforts  of  Peirce,  Chauvenet,  and  their  followers.  It  is  significant  that 
Loomis,  in  a  late  edition  of  his  trigonometry,  has  been  driven  by  the 
demands  of  the  times  to  abandon  the  old  system. 

The  advisability  of  using  infinitesimals  anil  the  idea  of  direction  in 
elementary  text-books  will  be  discussed  in  another  place. 

About  the  beginning  of  the  second  quarter  of  this  century  consider- 
able dissatisfaction  came  to  exist  among  the  j)ublic  about  the  college 
system  as  it  was  then  conducted  in  this  country.  The  people  demanded 
a  change  from  the  old  scholastic  methods.  Then  for  the  first  time  arose 
the  now  familiar  cry  against  forcing  the  ancient  languages  upon  all 
students  entering  college.  It  was  demanded  that  greater  prominence 
be  given  to  modern  languages,  to  English  literature,  to  practical  ine- 
chanics,  and  that  the  student  should  have  some  freedom  in  the  selection 
of  his  studies.  Though  some  few  modifications  were  made  here  and 
there  in  the  college  courses,  the  "  lifew  Education  "  did  not  secure  a 
firm  hold  upon  our  colleges  until  the  third  quarter  of  the  present 
century. 

In  these  reforms  Harvard  has  always  taken  a  prominent  part.  The 
elective  system  there  has  been  traced  back  to  1824,  when  Juniors  could 
choose  a  substitute  for  38  lessons  in  Hebrew,  and  Seniors  had  the 
choice  between  chemistry  and  fluxions.  Benjamin  Peirce  was  an  en- 
thusiastic advocate  of  the  elective  system. 

We  now  proceed  to  give  the  courses  in  mathematics  during  the  early 


136  TEACHING   AND    HISTOEY   OF   MATHEMATICS. 

part  of  Peirce's  connection  with  the  college.  His  own  text-books  were 
adopted  as  soon  as  they  came  from  the  press.  In  1836  and  1837  the 
Preshmenused  Walker's  Geometry,  Smyth's  Algebra,  Peirce's  Plane  and 
Spherical  Trigonometry  5  the  Sophomores,  Farrar's  books  on  Analytical 
Geometry,  Calculus,  and  Natural  Philosophy  5  the  Juniors  continued  the 
Natural  Philosophy. 

In  the  catalogue  for  1838  we  notice  important  changes.  The  Freshmen 
studied  Peirce's  Geometry  and  Algebra ;  the  Sophomore  class  was  di- 
vided into  three  sections,  of  which  the  first  pursued  practical  mathe- 
matics, including  mensuration,  dialing,  construction  of  charts,  survey- 
ing, the  use  of  globes  and  instruments  in  surveying,  during  the  first 
term  ;  and  during  the  second  term  the  general  principles  of  civnl  engi- 
neering, nautical  astronomy,  and  the  use  of  the  quadrant.  This  section 
was  evidently  intended  to  meet  the  demands  of  the  time  for  practical 
knowledge,  without  having  first  laid  a  broad  and  secure  theoretical 
foundation.  But  little  could  be  accomplished  in  civil  engineering  with- 
out a  knowledge  of  calculus.  The  second  section  reviewed  arith- 
metic, geometry,  and  algebra;  then  took  up  conic  sections,  fluxions,  and 
the  mathematical  theory  of  mechanics.  The  third  section,  intended  for 
students  of  mathematical  talents  and  taste,  pursued  analytic  geometry, 
theory  of  numbers  and  functions,  difierential  and  integral  calculus,  and 
mechanics. 

But  this  arrangement  did  not  prove  satisfactory.  The  facts  are  that 
Professor  Peirce's  text-books  were  found  very  difficult,  and  Peirce  him- 
self was  not  a  good  teacher,  except  for  boys  of  mathematical  genius. 
Peirce  was  anxious  to  introduce  the  elective  system,  so  that  students 
without  mathematical  ability  would  not  be  forced  to  pursue  mathematics 
beyond  their  elements.  In  May,  1838,  a  vote  was  passed,  permit- 
ting students  to  discontinue  their  mathematics  at  the  end  of  the  Fresh- 
man year  if  they  chose  to.  The  catalogue  for  1839  announced  that 
*' every  student  who  has  completed  during  the  Freshman  year  the 
studies  of  geometry,  and  algebra,  plain  trigonometry  with  its  applica- 
tions to  heights  and  distances,  to  navigation,  to  surveying,  and  that  of 
spherical  trigonometry,  and  who  has  passed  a  satisfactory  examination 
in  each  to  the  acceptance  of  the  mathematical  department  and  a  commit- 
tee of  the  overseers — may  discontinue  the  study  of  mathematics  at  the 
end  of  the  Freshman  year,  at  the  written  request  of  his  parent." 

Eeferring  to  these  changes  the  president  said,  the  following  year,  that 
the  liberty  to  discontinue  mathematics  at  the  end  of  the  year  had  been 
found  highly  acceptable  to  both  students  and  parents  and  had,  thus  far, 
been  attended  by  no  ill  consequences ;  that  elections  in  the  secondary 
course  had  had  a  tendency  to  encourage  those  capable  of  profiting  by 
the  study  of  that  branch;  thatthose  possessing  mathematical  talentwere 
stimulated;  that  of  fifty-five,  only  eight  continued  mathematics;  and 
that  the  head  of  the  department  considered  the  voluntary  system 
superior.    The  difdculties  in  the  mathematical  course  for  the  Sopho- 


INFLUX  OF  FRENCH  MATHEMATICS. 


137 


mores  seemed  to  be  removed.  But  how  about  the  Freshmen  ?  Mathe- 
matical studies  were  not  popular  with  them ;  they  complained  of  over- 
work. In  1839  the  committee  on  studies  reported  that  "  the  mathe- 
matical studies  of  the  Freshman  class  are  so  extensive  as  to  encroach 
materially"  upon  the  time  and  attention  due  to  other  branches,"  and  pro- 
posed to  remove  the  time  when  mathematical  studies  may  be  discon- 
tinued, from  the  end  of  the  Freshman  to  the  first  term  of  the  Sopho- 
more year. 

The  catalogue  for  1838-39  gives  no  mathematics  for  the  Junior  and 
Senior  years.  The  following  year  Peirce's  Treatise  on  Sound  was  studied 
by  the  Juniors.  In  1841  an  extended  mathematical  course  was  of- 
fered in  the  Junior  and  Senior  years.  The  Juniors  were  to  study  Peirce's 
Treatise  on  Sound  and  the  Calculas  of  Variations  and  Eesiduals ;  the 
Seniors,  Poisson's  M6canique  Analytique  and  Celestial  Mechanics. 
The  number  of  students  venturing  to  enter  such  difficult  but  enchanted 
fields  of  study  were  but  few.  In  1843  there  were  only  two  sections  in- 
stead of  three  as  before.  One  was  called  the  course  in  Practical  Mathe- 
matics, comprising  Peirce's  Plane  and  Spherical  Trigonometry  j  the 
other  was  called  the  course  in  Theoretical  Mathematics,  in  which  Peirce's 
Algebra  was  concluded,  and  his  Curves,  Functions,  and  Forces,  studied 
as  far  as  "  Quadratic  Loci."  These  two  courses  continued  through  the 
Junior  and  Senior  years.  The  studies  offered  varied  somewhat  from 
year  to  year. 

In  obedience  to  the  practical  demands  of  the  times,  the  Lawrence 
Scientific  School  was  opened  in  1842  as  a  branch  of  Harvard.  It 
began  as  a  school  of  chemistry.  But  by  the  year  1847  the  plan  of  this 
school  was  broadened  so  as  to  embrace  other  sciences.  "  There  shall 
be  established  in  the  University  an  advanced  school  for  instruction  in 
theoretical  and  practical  science  and  in  other  usual  branches  of  academic 
learning."  Instruction  was  to  be  given  by  Professor  Horsford  in  chem- 
istry, by  Professor  Agassiz  in  zoology  and  geology,  by  Professor  Lov- 
ering  in  experimental  philosophy,  by  William  Bond  in  practical  astron- 
omy, and  by  Professor  Peirce  in  higher  mathematics,  especially  in 
analytical  and  celestial  mechanics.  The  course  oft'ered  by  Professor 
Peirce  to  students  in  this  school,  in  1848,  was  as  follows : 

Course  in  Matueaiatics  and  Astronomy, 

i.— curves  an1>  functions. 


Regular  course. 

Peirce.  Curves  and  Functions. 

La  Croix.  CalculDiif6rential  et  Integral. 

Cauchy.  Les  Applications  du  Calcul  In- 
finitesimal h  la  Geoni6trie. 

MoNGE.  Application  do  1' Analyse  d  la 
G6ometrie. 


Parallel  covroe. 

BiOT.  G6ora6tT\e  Analytique. 

Cauchy.  Coups  d'Analyse  de  I'jGcole  Ro- 

yale  Polytechuique. 
Hamilton's  researches  respecting    qua 

ternions.     (Transactions  of  the  Royal 

Irish  Academy,  Vol.  XXI.) 


138 


TEACHING   AND   HISTOEY   OF   MATHEMATICS. 


II. — ANALYTICAL  AND  CELESTIAL  MECHANICS. 

Regular  course.  I  Farellel  course. 


Laplace.  M^canique  Celeste,  translated, 

with  a  Commentary,  by  Dr.  Bowditcii. 

Vol.  I. 
BOWDITCH.  On  the  Computation  of  the 

Orbits  of  a  Planet  or  Comet ;  Appendix 

to  Vol.  Ill  of  his  translation. 
Airy.  Figure  of  the  Earth,  from  the  En- 

cgclopcedia  Metropolitana. 
AiitY.  Tides, fvomthe£nci/olo^wdia Metro- 
politana. 


PoissoN.  MiScanique  Analytique. 

Lagrange.  M^canique  Analytique. 

Hamilton.  General  Method  in  Dynamics, 
from  the  London  Philosophical  TranS' 
actions  for  1834  and  1835. 

Gauss.  Theoria  Motus  Corporum  Cceles" 
tium. 

Bessel.  Untersuchungen. 

Leverriee.  D^veloppements  sur  Plu- 
sieurs  Points  de  la  Th^orie  des  Pertur- 
bations des  Plan^tes. 

Leverrier.  Les  Variations  S^culaires 
des  ]2l6mens  des  Orbites,  pour  les  Sept 
Planetes  Principales. 

Leverrier.  Th6orie  des  Mouvements  de 
Mercure. 

Leverrier.  Recherches  sur  les  Mouve- 
ments  de  la  Planfete  Herschel. 

Adams.  Explanation  of  the  Observed  Ir- 
regularities in  the  Motion  of  Uranus,  on 
the  Hypothesis  of  Disturbances  caused 
by  a  more  distant  Planet. 


III.— MECHANICAL  THEORY  OF  LIGHT. 
Regular  course.  Parallel  course. 


Airy.  Mathematical  Essays. 

MacCullagh,  On  the  Laws  of  Crystal- 
line Eeflection  and  Refraction.  (Trans- 
actions of  the  Royal  Irish  Academy,Vol. 
XVIII.) 


Catjchy.  Exercices  d' Analyse  et  de  Phys- 
ique Math^matiqnes. 

Neumann.  Theoretische  Untersuchung 
der  Gesetze,  nach  Welchen  das  Licht 
reflectirt  und  gebrochen  wird.  (Trans- 
actions of  Berlin  Academy  for  1835.) 


Sucli  a  course  of  studies  had  never  before  been  open  to  American 
students  in  any  American  college.  "Such  a  course,  or  any  other  equally 
advanced,  was  never  presented  in  any  other  American  institution  be- 
fore the  arrival  at  the  Johns  Hopkins  University  of  Professor  Sylvester. 
It  must  be  admitted  that  the  great  mass  of  Harvard  students  never 
studied  more  mathematics  than  was  absolutely  required  for  their  degree, 
but  now  and  then  Peirce  had  a  pupil  who  liked  mathematics,  under- 
stood the  greatness  of  his  teaching,  and  appreciated  and  loved  his 
character.  Peirce  was  the  center  of  an  influence  which  led  to  the  start- 
ing of  many  a  since  distinguished  scientific  career.  Prof.  T.  H.  Safford, 
one  of  his  favorite  pupils,  says:  "  Among  distinguished  scholars  of  the 
years  which  I  remember,  were  Prof.  G.  P.  Bond,  afterward  of  the  Ob- 
servatory ;  Dr.  B.  A.  Gould,  celebrated  as  an  astronomer ;  Eev.  Thomas 
Hill,  for  a  while  president  of  Harvard ;  Prof.  J.  D.  Pamkle,  of  the  In- 
stitute of  Technology,  Boston ;  Prof.  J.  E.  Oliver,  of  Cornell  Univer- 


INFLUX   OP   FKENCH   MATHEMATICS.  139 

sity ;  Prof.  A.  Hall  and  Prof.  S.  Newcomb,  of  the  F.  S.  l^avy ;  Mr.  W. 
P.  G.  Bartlett,  since  deceased ;  Mr.  G.  W.  Hill,  of  the  Nautical  Al- 
manac Office;  Mr.  Ohauncey  Wright,  known  as  a  philosopher ;  Prof. 
James  M.  Peirce  and  President  Eliot,  of  Harvard  ;  Prof.  C.  M.  Wood- 
ward, of  St.  Louis ;  Rev.  G.  W.  Searle,  of  St.  Paul's  (R.  0.)  Church, 
New  York  City ;  Prof.  W.  Watson,  formerly  of  the  Institute  of  Tech- 
nology ;  Professor  Byerly,  now  at  Harvard."  * 

Of  Peirce  as  a  teacher.  Dr.  A.  P.  Peabody  gives  us  an  interesting 
account.!  It  refers  to  the  first  year  that  Peirce  was  professor  at  Har- 
vard. 

"  For  the  academic  year  1832-33, 1,  as  tutor,  divided  the  mathemat- 
ical instruction  with  Mr.  Peirce.  *  *  *  He  took  to  himself  the  in- 
struction of  the  Freshmen.  The  instruction  of  the  other  three  classes 
we  shared,  each  of  us  taking  two  of  the  four  sections  into  which  the 
class  was  divided,  and  interchanging  our  sections  every  fortnight. 
*  *  *  In  one  respect  I  was  Mr.  Peirce's  superior,  solely  because  I 
was  so  very  far  his  inferior.  I  am  certain  I  was  the  better  instructor 
of  the  two.  The  course  in  the  Sophomore  and  Junior  years,  embracing 
a  treatise  on  the  Differential  Calculus,  with  references  to  the  calculus  in 
the  text-books  on  mechanics  and  other  branches  of  mixed  mathematics, 
was  hardly  within  the  unaided  grasp  of  some  of  our  best  scholars;  and, 
though  no  student  dared  ro  go  to  the  tutor's  room  by  daylight,  it  was 
no  uncommon  thing  for  one  to  come  furtively  in  the  evening  to  ask  his 
teacher's  aid  in  some  difficult  problem  or  demonstration.  For  this  pur- 
pose resort  was  had  to  me  more  frequently  than  to  my  colleague,  and 
often  by  students  who  for  the  fortnight  belonged  to  one  of  his  sections. 
The  reason  was  obWous.  No  one  was  more  cordially  ready  than  he  to 
give  such  help  as  he  could;  but  his  intuition  of  the  whole  ground  was 
so  keen  and  comprehensive  that  he  could  not  take  cognizance  of  the 
slow  and  tentative  processes  of  mind  by  which  an  ordinary  learner  was 
compelled  to  make  his  step-by-step  progress.  In  his  explanations  he 
would  take  giant  strides;  and  his  frequent,  'you  see,'  indicated  what 

he  saw  clearly,  but  that  of  which  his  pupils  could  get  hardly  a  glimpse. 

m     #     * 

"  Our  year's  work  was  on  the  whole  satisfactory,  and  yet  I  think  that 
we  were  both  convinced  that  the  differential  calculus  ought  not  to  have 
been  a  part  of  a  prescribed  course.  There  was  a  great  deal  of  faltering 
and  floundering,  even  among  else  good  scholars.  *  *  *  Our  exam- 
inations were  viva  voce,  in  the  presence  of  a  committee  of  reputed  ex- 
perts in  each  several  department.  We  shrank  from  the  verdict  of  our 
special  committee  in  no  part  of  our  work  except  the  calculus.  As  the 
day  approached  for  the  examination  in  that  branch  we  were  solicitous 
that  Robert  Treat  Paine,  who  was  on  the  committee,  should  not  be 
present ;  for  we  supposed  him  to  be  the  only  member  of  the  committee 
who  was  conversant  with  the  calculus.    He  did  not  come,  and  we  were 

"Letter  to  the  writer,  November  6, 1888.         t  Harvard  Eeminiscences,  1888,  p.  182. 


140  TEACHING    AND    HISTORY    OF    MATHEMATICS. 

glad.  *  *  *  If  there  were  defects  and  sliortcomings,  there  was  cer- 
tainly no  one  present  who  could  detect  them." 

Peirce  had  no  success  in  teaching  mathematics  to  students  not  math- 
ematically inclined.  Sepoated  and  loud  complaints  were  made  at  Har- 
vard that  the  mathematical  teaching  was  poor.  The  majority  of  students 
disliked  the  study  and  dropped  it  as  soon  as  possible.  Says  Prof.  Will- 
iam F.  Allen  (class  of  1851)  in  a  letter  to  the  writer : 

"  I  am  no  mathematician,  but  that  I  am  so  little  of  one  is  due  to  the 
wretched  instruction  at  Harvard.  Professor  Peirce  was  admirable  for 
students  with  mathematical  minds,  but  had  no  capacity  with  others. 
He  took  only  elective  classes,  and  of  course  I  didn't  elect.  Only  two 
did  in  our  class  of  about  sixty,  and  I  believe  they  soon  dropped  it.  lii 
my  Freshman  year  I  had  very  good  instruction  from  Mr.  Child,  now 
the  professor  of  English  literature,  and  editor  of  ballads.  I  had  alge- 
bra and  geometry  with  him,  and  did  fairly  well.  In  the  Sophomore 
year  (trigonometry  and  analytic  geometry)  we  had  a  different  instructor, 
and  it  was  a  mere  farce.  In  analytic  geometry  I  was  taken  up  once  in 
the  course  of  the  term,  on  rectangular  co-ordinates  in  space,  and  I 
knew  perfectly  well  (although  I  was  never  so  told)  that  at  examination  I 
should  be  called  up  upon  rectangular  co-ordinates  in  space.  (Written 
examinations  had  never  been  heard  of.)  When  examination  day  came 
(a  committee  in  attendance)  the  tutor  was  sick,  and  a  shudder  ran 
through  the  class.  But  he  heroically  pulled  himself  together  and  held 
his  examination  in  person,  and  I  was  examined  upon  rectangular  co- 
ordinates in  space.  The  sum  of  my  knowledge  of  analytical  geometry 
at  the  present  day  is  that  there  are  such  things  (or  were)  as  rectangular 
co-ordinates  in  space — and  I  suppose  there  must  also  be  some  out  of 
space.  *  *  *  Peirce's  text-books  were  used.  His  geometry  I  liked 
much,  also  the  algebra,  only  that  it  was  pretty  hard.     *    *     * 

''  I  graduated  in  1851,  and  I  remember  when  I  was  in  Germany  two  or 
three  years  later,  I  met  a  gentleman  who  had  just  returned  from  Amer- 
ica— a  young  German  Gelehrte — and  he  assured  me  that  there  was  not 
one  mathematician  in  the  United  States,  and  only  one  astronomer, 
Peirce.  It  was  not  an  agreeable  thing  to  be  told,  for  a  patriotic  young 
American  as  1  was  then,  but  I  suppose  it  was  not  far  from  the  truth." 

In  January,  1848,  Thomas  Sherwin,  by  order  of  the  committee  for 
examination  in  mathematics  reported  that  in  1847  there  were  present 
for  examination  but  one  Senior  in  Bowditch's  Laplace,  and  only  five 
Juniors  in  Curves  and  Functions.  He  went  on  to  say  that  mathe- 
matics could  be  made  attractive,  that,  hence,  arose  the  inquiry,  why 
this  study  was  so  very  decidedly  unpopular  at  the  University,  and 
why  so  general  an  opinion  prevailed  throughout  the  community,  that 
the  student  stood  less  creditable  in  this  branch  than  in  others.  The 
answer  to  this  was  that  the  text  books  were  abstract  and  difficult,  that 
few  could  comprehend  them  without  much  explanation,  that  Peirce's 
works  were  symmetrical  and  elegant,  and  could  be  perused  with  pleas- 


INFLUX  OF  FRENCH  MATHEMATICS.  141 

ure  by  the  adult  mind,  but  that  books  for  young  students  should  be  more 
simple.  The  report  then  says  that  there  are  mathematical  works  of  no 
small  merit,  which  embraced  the  same  subjects  as  the  text-books 
now  used,  which  were  much  less  difficult  of  comprehension,  such  as 
Bourdon's  Algebra,  J.  E.  Young's  Treatises,  and  a  recent  edition  of 
Button's  Mathematics. 

The  majority  report  was  followed  by  a  minority  report  by  Thomas 
Hill  and  J.  Grill,  which  differed  regarding  the  text-books  at  Harvard. 
'•  Your  minority  of  the  committee  believe  that  these  text-books,  by  their 
beauty  and  compactness  of  symbols,  by  their  terseness  and  simplicity  of 
style,  by  their  vigor  and  originality  of  thought,  and  by  their  happy 
selection  of  lines  of  investigation,  offer  to  the  student  a  beautiful  model 
of  mathematical  reasoning,  and  lead  him  by  the  most  direct  route  to  the 
higher  regions  of  the  calculus.  For  those  students  who  intend  to  go  far- 
ther than  the  every- day  applications  of  trigonometry,  this  series  of  books 
is,  in  the  judgment  of  the  minority,  by  far  the  best  series  now  in  use." 

While  the  good  qualities  of  Peirce's  text-books,  as  described  by  the 
minority,  must  be  acknowledged,  it  is  nevertheless  true,  that  owing  to 
their  compactness  and  brevity,  which  characterize  all  the  writings  of 
Peirce,  the  books  seemed  obscure  to  beginners.*  Still,  however,  they 
continued  to  be  used  at  Harvard  for  many  years  longer. 

Professor  Peirce  said  in  his  report  of  I^ovember  6, 1849,  on  the  teach- 
ing of  higher  mathematics  in  the  college  and  the  Lawrence  Scientific 
School,  that  he  had  two  pupils.  One  of  these  students  was  a  member 
of  the  Lawrence  Scientific  School,  and  the  other  was  the  child,  T.  Henry 
Safford,  who  had  attracted  so  much  attention  for  his  early  development 
of  mathematical  ability.  "  These  two  students  attended  lectures  on 
analytical  mechanics,  and  young  Salford  showed  himself  perfectly  com- 
petent to  master  the  difficult  subject  of  research,  and  once  or  twice  sur- 
prised his  teacher  by  the  readiness  with  which  he  anticipated  the  ob- 
ject of  some  peculiar  form  of  transformation.  Up  to  this  time  Safford 
fully  realizes  his  early  promise  of  extraordinary  powers  as  a  geometer, 
but  his  friends  cannot  free  themselves  from  apprehension,  when  they 
perceive  that  the  growth  of  his  body  does  not  correspond  to  that  of  his 
intellect."  He  then  states  that  with  the  mathematical  pupil  of  the 
school  the  professor  read  also  Lagrange's  Mecanique  Analytiqtie  and  La- 
place's Theorie  Analytique  des  Frobabilites. 


*In  another  place,  Rev.  Thomas  Hill  speaks  of  these  books  as  follows  (memorial  col- 
lection by  Moses  King)  :  ''His  test-books  were  also  complained  of  for  their  condensa- 
tion, as  being  therefore  obscure  ;  but  under  competent  teachers,  the  brevity  was  the 
cause  of  their  superior  lucidity.  lu  the  Waltham  High  School  his  books  were  used 
for  many  years,  and  the  graduates  attained  thereby  a  clear  and  more  useful,  applicable 
knowledge  of  mathematics  than  was  given  at  any  other  high  school  in  this  country; 
nor  did  they  find  any  difficulty  in  mastering  even  the  demonstration  of  Arbogast's  Poly- 
nomial Theorem,  as  presented  by  Peirce.  The  latter  half  of  tlie  volume  on  the  inte- 
gral calculus,  full  of  the  marks  of  a  great  analytical  genius,  is  the  only  part  of  all 
his  text-books  really  too  difficult  for  students  of  average  ability.'"' 


142  TEACHING   AND    HISTOKY    OF    MATHEMATICS. 

As  regards  the  number  of  students  electing  mathematics,  the  com- 
mittee of  overseers  stated,  in  1849,  that  as  long  as  the  choice  is  offered, 
the  lighter  labor  will  always  be  preferred,  and  that  this  tendency  will 
probably  get  stronger.  "  Hebrew  roots  and  jiolynomial  roots  will  be 
neglected  in  a  garden  abounding  with  French  bouquets  and  Italian 
musicj  and  even  now  it  can  not  surprise  us  that,  while  the  Smith  pro- 
fessor of  French  and  Spanish  language  and  literature,  and  instructor 
in  Italian,  is  surrounded  by  a  gay  crowd  of  utilitarian  admirers,  the 
Perkins  professor  of  mathematics  and  astronomy  is  working  in  his  deep 
mines  for  one  infant  prodigy  and  one  eminent  Senior."  Some  Juniors 
studied  analytical  statics,  and  gave  the  best  evidence  of  successful  devo- 
tion to  the  subject. 

The  elective  system  was  abandoned  almost  completely  in  1850.  Mathe- 
matical studies  were  elective  only  in  the  Junior  and  Senior  years.  In 
1867  the  elective  system  was  again  adopted  at  Harvard,  and  on  the 
most  liberal  scale.  Sophomore  mathematics  were  again  no  longer  "  re- 
quired." Peirce's  books  still  held  their  ground.  The  only  invaders 
were  J.  M.  Peirce's  Analytical  Geometry,  and,  in  1865,  Puckle's  Analytic 
Geometry.  In  1869  the  committee  of  overseers  reported  that  mathe- 
matics was  a  required  study  only  for  the  Freshmen;  that  elective  mathe- 
matics were  taken  this  year  by  one  hundred  Sophomores,  six  Juniors, 
and  eight  Seniors ;  that  the  Sophomores  and  Juniors  could  elect  either 
pure  or  applied.  They  also  stated  that  "the  number  electing  this  de- 
partment in  the  upper  classes  is  never  expected  to  be  large,  as  the 
studies  are  advanced  beyond  what  most  students  have  either  aptitude 
or  occasion  for." 

We  find  that,  "for  the  last  few  years  of  his  life  Professor  Peirce  had 
for  his  pupils  only  young  men  who  were  prepared  for  profounder 
study  than  ever  entered  into  a  required  course,  or  a  regularly  planned 
curriculum ;  but  he  never  before  taught  so  efficiently,  or  with  results 
so  worthy  of  the  mind  and  heart  and  soul,  which  he  always  put  into 
his  work."  * 

It  will  be  instructive  to  listen  to  what  former  pupils  of  Peirce  have 
to  say  of  him.  Prof.  Truman  Henry  Saftbrd,  of  Williams  College,  says 
in  a  letter  to  the  writer :  "  I  was  a  student  at  Harvard  in  the  class  of 
1854.  Prof.  B.  Peirce  taught  the  Sophomores,  I  believe  (I  entered  the 
Junior  class),  but  not  very  well;  he  had  hardly  patience  enough,  I  sup- 
pose. To  the  Juniors  and  Seniors  he  lectured  on  higher  algebra,  the 
calculus,  and  analytical  mechanics.  His  lectures  were  substantially 
contained  in  his  text- books— Algebra,  Curves  and  Functions,  and  Ana- 
lytic Mechanics.  They  were  very  interesting  and  inspiring  to  those 
who  could  follow  them.  There  was  but  little  practice;  the  examples 
in  the  book  were  generally  worked  out.  In  my  class  a  number  (twelve 
or  so)  took  the  first  year's  work  ;  the  second,  which  included  integral 
calculus,  complex  numbers,  and  analytical  mechanics,  was  taken  by 

•  Harvard  Eeminiscences,  by  A.  P.  Peabody,  p.  186. 


INFLUX  OP  FRENCH  MATHEMATICS.  143 

four  only.  One  of  them  was  0.  K.  Lowell,  afterwards  a  cavalry  gen- 
eral in  the  Civil  War,  a  nephew  of  Professor  Lowell ;  another,  George 
Putnam,  Esq.j  a  third,  W.  C.  Paine,  afterwards  a  West  Point  scholar, 
where  he  was  first  in  his  class,  and  a  lieutenant  of  engineers,  but  he 
resigned  as  a  captain.  The  fourth  was  myself.  I  had  heard  some  of 
Professor  Peirce's  lectures  some  years  before,  while  a  school-boy,  but 
could  not  follow  them  so  well." 

For  some  years  following  1838  Prof.  Joseph  Lovering  taught  classes 
in  mathematics.  Of  him  and  Peirce,  Edward  E.  Hale  says :  *  "  The 
classical  men  made  us  hate  Latin  and  Greek  5  but  the  mathematical 
men  (such  men !  Peirce  and  Lovering)  made  us  love  mathematics,  and 
we  shall  always  be  grateful  to  them." 

Says  Thomas  Wentworth  Higginson:  "As  to  mathematical  instruc- 
tion, this  reform  (elective  system)  was  an  especial  benefit,  for  Professor 
Peirce's  genius  revelled  in  the  new  sensation  of  having  voluntary  pupils, 
and  he  gave  a  few  of  us  his  Curves  and  Functions  as  lectures,  with  run- 
ning elucidations.  Nothing  could  be  more  stimulating  than  to  see  our 
ardent  instructor,  suddenly  seized  with  a  new  thought  and  forgetting 
our  very  existence,  work  away  rapidly  with  the  chalk  upon  a  wholly 
new  series  of  equations  5  and  then,  when  he  had  forced  himself  into  the 
utmost  corner  of  the  blackboard  and  could  get  no  farther,  to  see  him 
come  back  to  earth  with  a  sigh  and  proceed  with  his  lecture.  We  did 
not  know  whither  he  was  going,  but  that  huddle  of  new  equations 
seemed  like  a  sudden  outlet  from  this  world,  and  a  ladder  to  the  stars. 
He  gave  a  charm  to  the  study  of  mathematics  which  for  me  has  never 
waned,  although  the  other  pursuits  of  life  soon  drew  me  from  that  early 
love.  This  I  have  always  regretted,  and  so  did  Peirce,  who  fancied  that 
I  had  some  faculty  that  way,  and  had  me  put,  when  but  eighteen,  on  a 
committee  to  examine  the  mathematical  classes  of  the  college.  Long 
after,  when  I  was  indicted  for  the  attempted  rescue  of  a  fugitive  slave, 
and  the  prison  walls  seemed  impending,  I  met  him  in  the  street  and  told 
him  that  if  I  were  imprisoned  I  should  have  time  to  read  Laplace's 
Mecanique  Celeste.  *In  that  case,'  said  the  professor,  who  abhorred  the 
abolitionists,  '  I  sincerely  wish  you  may  be.' " 

Among  the  more  prominent  mathematical  tutors  of  this  period  may 
be  mentioned  0.  W.  Eliot,  now  president  of  Harvard,  and  James  Mills 
Peirce,  a  son  of  Benjamin  Peirce.  The  latter  graduated  in  1853,  was 
tutor  from  1854  to  1858,  and  from  1860  to  1861,  when  he  was  made  as- 
sistant professor  of  mathematics.  In  1867  he  became  university  profes- 
sor of  mathematics. 

Benjamin  Peirce  presided  for  some  years  over  a  mathematical  society. 
It  comprised  eight  or  ten  men  of  some  reputation  in  Boston  and  Cam- 
bridge, who  met  to  discuss  mathematical  topics.  Each  member  would 
present  to  the  society  such  novelties  as  his  inquiries  into  some  particu- 
lar branch  had  suggested,  and  "  in  the  discussion  which  followed,  it 

*  How  I  was  Educated,  Fornm,  I,  April,  1886,  p.  61. 


144  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

would  almost  invariably  appear  that  Peirce  had,  while  the  paper  was 
being  read,  pushed  out  the  author's  methods  to  far  wider  results  than 
the  author  had  dreamed.*  His  mind  moved  with  great  rapidity,  and 
it  was  with  great  difficulty  that  he  brought  himself  to  writing  out  even 
the  briefest  record  of  its  excursions." 

We  now  proceed  to  a  brief  account  of  Benjamin  Peirce's  original  re- 
searches. Several  original  articles  were  contributed  by  him  to  the 
Mathematical  Miscellany  and  to  the  Gambridge  Miscellany.  Peirce  had 
planned  an  extended  treatise  on  Physical  and  Celestial  Mechanics,  to 
be  developed  in  four  systems,  of  Analytical  Mechanics,  Celestial  Me- 
chanics, Potential  Physics,  and  Analytic  Morphology.  Of  these  four, 
only  one  appeared,  the  system  of  Analytic  Mechanics,  in  1855.  The  sub- 
stance of  this  was  prepared  as  a  part  of  a  course  of  lectures  for  math- 
ematical students  at  Harvard.  The  publication  was  undertaken  at 
the  request  of  some  of  his  pupils,  especially  of  J.  D.  Eunkle.  He 
consolidated  the  latest  researches  into  a  consistent  and  uniform  treatise, 
and  carried  "  back  the  fundamental  principles  of  the  science  to  a  more 
profound  and  central  origin."  It  was  very  far  from  being  a  mere  com- 
pilation. Ih  his  books  he  supplanted  many  a  traditional  method  in 
mathematics  by  concise  and  axiomatic  definitions  and  demonstrations 
of  his  own  invention.  As  an  instance  of  this  we  mention  his  assump- 
tion as  self-evident  that  a  line  which  is  wholly  contained  upon  a  limited 
surface,  but  which  has  neither  beginning  nor  end  on  that  surface,  must 
be  a  curve  re-entering  upon  itself.  By  this  new  axiom  he  reduces  a 
demonstration  which  would  otherwise  occupy  half  a  dozen  pages  to  a 
few  lines. t 

Peirce's  Analytical  Mechanics  was  generally  acknowledged  at  the  time, 
even  in  Germany,  to  be  the  best  of  its  kind.  |  An  American  student  in 
Germany  asked  once  an  eminent  German  professor  what  book  he  would 
recommend  on  analytical  mechanics.  The  reply  was  instantaneous, 
"  There  is  nothing  fresher  and  nothing  more  valuable  than  your  own 
Peirce's  recent  quarto." 

Benjamin  Peirce  was  much  interested  in  the  comet  of  1843,  and  in  a 
few  lectures  he  aroused  by  his  great  eloquence  an  interest  in  astronomy 
which  led  to  the  foundation  of  the  observatory  of  Cambridge.  His 
mathematical  ability  was  first  brought  into  general  notice  in  con- 
nection with  the  discovery  of  Neptune.  Messrs.  Adams,  of  Gambridge, 
and  Leverrier,  of  Paris,  had  calculated,  from  theory  alone,  where  this 
planet  ought  to  appear  in  the  heavens,  If  visible,  and  Galle,  of  Berlin, 
discovered  on  September  23,  1846,  the  planet  at  the  place  indicated  to 
him  by  Leverrier.  Peirce  began  to  study  the  planet's  motion,  and  came 
to  the  conclusion  that  its  discovery  was  a  happy  accident;  not  that 
Leverrier's  calculations  had  not  been  exact,  and  wonderfully  laborious, 

*  Nation,  October  14,  1880. 

t  Rev.  Thomas  Hill,  in  the  Memorial  Collection,  by  Moses  King. 

i  Nature.    October  2S,  1880. 


INFLUX  OF  FKENCH  MATHEMATICS.  145 

and  deserving  of  the  bigliest  honor,  but  because  there  were,  in  fact, 
two  very  different  solutions  of  the  perturbations  of  Uranus  possible; 
Leverrier  had  correctly  calculated  one,  but  the  actual  planet  in  the  sky 
represented  the  other,  and  the  actual  j)Ianet  and  Leverrier's  ideal  one 
lay  in  the  same  direction  from  the  earth  only  in  1846. 

Astronomers  of  to-day  would  hardly  accept  Peirce's  conclusions. 
"His  views  came,  probably,  from  a  misapprehension  of  Leverrier's 
methods.  There  are  two  methods  by  which,  in  theory,  the  problem 
could  be  approached,  that  of  general  and  that  of  special  perturbations. 
Leverrier  used  the  latter,  while  Peirce's  criticisms  seem  directed  against 
the  former."  * 

On  February  2,  1847,  Mr.  C,  Walker,  of  Washington,  discovered  that 
a  star  observed  by  Lalande  in  May,  1795,  must  have  been  the  planet 
ISTeptune.  This  observation  afforded  the  means  of  an  accurate  deter- 
mination of  the  orbit.  Walker's  orbit  of  Neptune  furnished  Peirce 
with  materials  for  still  more  thorough  investigation  of  the  theory  and 
re-determination  of  the  perturbations.  These  perturbations  enabled 
Walker  to  get  an  orbit  more  correct,  which  Peirce  used  again  in  his 
turn.  Thus,  eighteen  months  after  the  discovery  of  Neptune  its  orbit 
was  calculated  by  American  astronomers  so  accurately  that  the  con- 
formity between  the  predicted  and  observed  places  was  far  more  close 
for  Neptune  than  any  other  planet  in  the  heavens. t 

A  feW;  years  .later  Peirce  published  his  investigations  on  Saturn's 
rings.  The  younger  Bond  had  seen  the  ring  divide  itself  and  re-unite, 
and  had  been  led  by  this  to  deny  the  solidity  of  their  structure. .  Peirce 
followed  with  a  demonstration,  on  abstract  grounds,  of  their  non-sol- 
idity.|  The  same  subject  was  afterward  investigated  again  in  England 
by  James  Clerk  Maxwell. 

Admiral  C.  H.  Davis,  a  relative  of  Peirce,  succeeded  in  persuading 
Congress  to  pay  for  the  calculation  of  an  American  almanac  for  the 
sailors,  so  that  we  would  not  be  dependent  upon  foreigners,  which 
might  be  troublesome  in  case  of  war.  The  Nautical  Almanac  Office 
was  established  at  first  in  Cambridge,  under  Davis's  business  manage- 
ment and  Peirce's  scientific  control. §  One  of  the  assistants  in  the 
office,  appointed  in  1849,  was  J.  D.  Runkle,  then  one  of  Peirce's  pupils 
in  the  Lawrence  Scientific  School.  He  helped  in  the  preparation  of  the 
American  Ephemeris  and  Nautical  Almanac,  in  which  he  continued  to 
engage  till  1884. 

The  publications  of  this  office  gained  scientific  recognition  from  the 
first.    In  f  852  were  printed  Peirce's  Lunar  Tables,  to  be  used  in  making 

*Prof.  G.  C.  Comstock,  "Washburn  Observatory,  in  a  letter  to  the  writer.  Prof. 
C.  A.  Young  claims  that  the  discovery  was  not  an  accident  {General  Astronomy,  ip. 
371).  / 

t  Proceedings  American  Association  for  Advancement  of  Science,  Vol.  VIII,  1854, 
address  by  B.  A.  Gould,  jr.,  p.  18. 

t  Astronomical  Journal  (Gould's),  Vol.  II,  p.  5. 

$  Development  of  Astronomy  in  the  United  States,  by  T.  H.  Safford,  1888,  p.  21. 

881~No.  3 10 


146  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

computations  for  the  Nautical  Almanac.  They  were  intended  to  serve 
only  a  temporary  purpose  until  Hansen's  long  expected  tables  should 
make  tbeir  appearance,  but  they  continued  to  be  used  after  that.  He 
made  very  laborious  and  exact  calculations  of  the  occultations  of  the 
Pleiades,  which  furnished  means  of  studying  the  form  both  of  the  earth 
and  the  moon. 

From  1852  to  1867  Peirce  had  the  direction  of  the  longitude  observa- 
tions for  the  U.  S.  Coast  Survey,  and  in  1867,  after  the  death  of  Bache, 
he  was  appointed  Superintendent  of  the  U.  S.  Coast  Survey,  which  office 
he  held  till  1874. 

Benjamin  Peirce  was  more  celebrated,  in  his  day,  as  a  mathematical 
astronomer  than  as  a  cultivator  ol  ijure  mathematics.  His  most  im- 
portant researches  in  pure  mathematics  were  not  placed  in  reach  of  the 
mathematical  public  until  after  his  death.  In  our  opinion,  Peirce  will 
be  remembered  by  future  generations  for  his  investigations  on  Linear 
Associative  Algebra,  quite  as  well  as  for  his  other  scientific  achieve- 
ments. He  will  be  remembered  as  an  algebraist  as  well  as  an  astrono- 
mer. His  thoughts  were  turned  especially  toward  the  logic  of  mathe- 
matics and  the  limits  and  extension  of  fundamental  processes.  He  read 
several  papers  on  algebra  before  the  American  Academy  for  the  Advance- 
ment of  Science.  In  1870  one  hundred  lithographed  copies  of  a  memoir 
on  Linear  Associative  Algebra,  read  before  the  National  Academy  of 
Sciences,  were  taken,  for  distribution  among  his  friends.  This  memoir 
was  at  last  published  iu  the  American  Journal  of  Mathematics,  Vol. 
IV,  No.  2,  with  notes  and  addenda  by  C.  S.  Peirce,  son  of  the  author. 
Benjamiu  Peirce  himself  considered  this  memoir  the  best  of  his  scientific 
efforts.  The  lithographed  copies  contain  the  following  modest  intro- 
ductory remarks  by  the  author,  which  are  omitted  in  the  American 
Journal  of  Mathematics : 

"  To  MY  Friends  : " 

"  This  work  has  been  the  pleasantest  mathematical  effort  of  my  life. 
In  no  other  have  I  seemed  to  myself  to  have  received  so  full  a  reward 
for  my  mental  labor  in  the  novelty  and  breadth  of  the  results.  I  pre- 
sume that  to  the  uninitiated  the  formulas  will  appear  cold  and  cheer- 
less, but  let  it  be  remembered  that,  like  other  mathematical  formulse, 
they  find  their  origin  in  the  divine  source  of  all  geometry.  Whether  I 
shall  have  the  satisfaction  of  taking  part  in  their  exposition,  or  whether 
that  will  remain  for  some  more  profound  expositor,  will  be  seen  in  the 
future."*  • 

*  Peirce  distinguishes  his  algebras  from  each  other  by  the  number  of  their  funda- 
mental conceptions,  or  of  the  letters  of  their  alphabet.  Thus,  an  algebra  which  has 
only  one  letter  in  the  alphabet  is  a  single  algebra ;  one  that  has  two  a  double  algebra, 
and  80  on.  His  investigation  does  not  usually  extend  beyond  the  sextuple  algebra. 
This  classification  he  calls  "cold  and  uniustructive,  like  the  artificial  Linnaean  sys- 
tem of  botany."  "  But  it  is  useful  in  a  jiroliminary  investigation  of  algebras  until  a 
sufficient  variety  is  obtained  to  aiford  the  material  for  a  natural  classification."  He 
tihen  begins  his  researches  with  single  algebra,  then  goes  to  double  algebra,  and  so  on, 


INFLUX  OP  FKENCH  MATHEMATICS.  147 

Peirce's  memoir  is  a  wonderfal  volume.  It  is  almost  entitled  to  rank 
"  as  a  Principia  of  the  philosophical  study  of  the  laws  of  algebraical 
operation." 

One  of  the  pall-bearers  at  the  funeral  of  the  greatest  American  alge- 
braist was  Prof.  J.  J.  Sylvester. 

During  the  last  tea  years  of  his  life  Benjamin  Peirce  was  relieved 
of  much  of  the  labor  and  responsibility  falling  upon  the  head  of  a  de- 
partment iu  a  university  by  his  son,  Prof.  James  Mills  Peirce.  Though 
not  the  heir  of  his  father's  genius,  Prof.  J.  M.  Peirce  is  a  thorough  and 
able  mathematician.  He  excels  his  father  in  being  an  excellent  teacher. 
In  1857  he  ])ublished  an  Analytic  Geometry,  which  was  used  for  some 
years  as  a  text-book  at  Harvard.  He  has  also  published  Three  and  Four 
Place  Tables  of  Logarithmic  and  Trigonometric  Functions,  1871;  Ele- 
ments of  Logarithms,  1873,  and  Mathematical  Tables,  chiefly  to  Four 
Figures,  1st  Series,  1879. 

Connected  with  the  mathematical  department  are,  since  1870,  Prof. 
0.  .L  White;  since  1876,  Prof.  W.  E.  Byerly;  since  1881,  Prof.  Benja- 
min O.  Peirce,  and  Mr.  George  W.  Sawin. 

Professor  Byerly  published  in  1880  his  Elements  of  the  Differential 
Calculus,  and  in  1882  his  Elements  of  the  Integral  Calculus.  Byerly's 
Calculus  is  a  scholarly  work.  In  the  rigorous  treatment  and  judicious 
selection  of  subjects  and  adaptability  to  class  use  ic  is,  we  believe, 
surpassed  by  no  other  American  work.  Professor  Byerly  uses  the 
notation,  D^^y,  which  was  first  employed  in  this  country  by  Benjamin 
Peirce.  In  answer  to  a  letter  of  inquiry  regarding  the  history  of  this 
notation  Professor  Byerly  says  :*  "  It  was  certainly  used  with  some 

up  to  sextuple,  making  nearly  a  hundred  algebras,  which  he  shows  to  be  possible. 
Of  all  these,  only  three  algebras  had  ever  been  heard  of  before.  Of  the  two  single 
algebras  we  have  one — the  common  algebra,  including  arithmetic.  Of  the  three 
double  algebras  we  have  one,  the  calculus  of  Leibnitz  and  Newton.  Of  over  twenty 
quadruple  algebras  we  have  the  quaternions  of  Hamilton. 

Prof,  Arthur  Cayley,  in  his  presidential  address  before  the  British  Association,  in 
1883,  speaking  of  Peirce's  Linear  Associative  Algebra,  said :  "  We  here  consider  sym- 
bols A,  B,  etc.,  which  are  linear  functions  of  a  determinate  number  of  letters  or  units, 
i,  j,  Jc,  I,  etc.,  with  co-efficients  which  are  ordinary  analytical  magnitudes,  real  or  im- 
aginary (viz,  the  co-efficients  are  in  general  of  the  form  x  +  iy,  where  i  is  the  before- 
mentioned  imaginary,  or  V — 1).  The  letters  i,  j,  etc.,  are  such  that  every  binary  com- 
bination i^,  ij,  ji,  etc.,  (the  ij  being  in  general  not  equal  to  ji)  is  equal  to  a  linear  func- 
tion of  the  letters,  but  under  the  restriction  of  satisfying  the  associative  law,  viz,  for 
each  combination  of  three  letters  i/.fc=i.jX-,  so  that  there  is  a  determinate  and  unique 
product  of  three  or  more  letters  ;  or,  what  is  the  same  thing,  the  laws  of  combination 
of  the  units  i,j,  Tc,  are  defined  by  a  multiplication  table  giving  the  values  of  i*,  ij,ji, 
etc. ;  the  original  units  may  be  replaced  by  linear  functions  of  these  units,  so  as  to 
give  rise,  for  the  units  finally  adopted,  to  a  multiplication  table  of  the  most  simple 
form  ;  and  it  is  very  remarkable  how  frequently  in  these  simplified  forms  we  have 
nilpotent  or  idempotent  symbols  (1^=0,  or  i'^=i,  as  the  case  may  be),  and.  symbols 
i,j,  such  that  ijz=ji=0  ;  and,  consequently,  how  simple  are  the  forms  of  the  multipli- 
cation tables  which  define  the  several  systems,  respeotively." 

*  Letter  to  the  writer,  December  27,  1888. 


148  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

freedom  in  England  and  on  the  Continent  in  the  first  half  of  this  century. 
It  is  given  in  Barlow's  Mathematical  Dictionary,  1814,  was  used  by 
Cauchy  about  1830,  by  Tortolini  in  1344,  by  Schlomilch  in  1846,  and  by 
Boole  and  Carmichael  somewhat  later,  and  each  of  the  authors  I  have 
mentioned  uses  the  symbol  as  if  it  were  a  familiar  one  and  without 
reference  to  its  history." 

Prof.  Benjamin  Osgood  Peirce  has  published  Elements  of  the  Theory 
of  the  Newtonian  Potential  Function,  1886. 

Since  1867  great  changes  have  been  made  in  the  requirements  for 
admission  to  Harvard  and  in  the  arrangement  of  the  mathematical 
courses.  Since  that  time  the  elective  system  has  been  in  operation  in 
full  force.    The  terms  for  admission  have  been  much  increased. 

From  the  selected  sheets  of  the  Harvard  University  Catalogue  for 
1888-89  we  take  the  following  regarding  the  requirements  for  admis- 
sion, omitting  whatever  has  no  bearing  on  mathematics  : 

Tlie  examinations  for  admission  embrace  two  classes  of  studies,  elementary  and 
advanced. 

The  elementary  studies  are  not  supposed  to  be  equivalent  to  one  another ;  Greek, 
Latin,  and  mathematics  have  much  greater  weight  in  the  examinations  than  any  of 
the  rest. 

The  advanced  studies  are  supposed  to  be  equivalent  in  regard  to  time  spent  upon 
them  at  school,  and  will  have  the  same  weight  in  the  examinations.  Each  of  the 
advanced  studies  is  taught  in  college  in  an  elective  course  (or  two  half-courses)  occu- 
pying three  hours  a  week  for  a  year  ;  and  the  standard  required  at  the  entrance  ex- 
aminations is  the  same  as  in  the  corresponding  college  courses. 

The  elementary  studies  are  prescribed  for  all  candidates,  except  under  the  condi" 
tions  named  below  (Paragraph  I) ;  and  every  candidate  is  further  required  to  present 
himself  for  examination  in  not  less  than  two  of  the  advanced  studies. 

I.  The  advanced  study  numbered  6  together  with  one  of  the  three  numbered  7  (see 
below  under  "advanced  studies  in  mathematics"),  8  (physics),  and  9  (chemistry), 
may  be  substituted  for  either  elementary  Greek  or  elementary  Latin. 
Elementary  Studies  in  Mathematics — 

(a)  Algebra,  through  quadratic  equations.  (The  requirement  in  algebra  embraces 
the  following  subjects:  Factors,  common  divisors  and  multiples,  fractions,  ratios, 
and  proportions;  negative  quantities  and  the  interpretation  of  negative  results; 
the  doctrine  of  exponents;  radicals  and  equations  involving  radicals;  the  binomial 
theorem  for  positive  integral  exponents  and  the  extraction  of  roots ;  putting  ques- 
tions into  equations,  and  the  reduction  of  equations ;  the  ordinary  methods  of  elimi- 
nation, and  the  solution  of  both  numerical  and  literal  equations  of  the  first  and  sec- 
ond degrees,  with  one  or  more  unknown  quantities,  and  of  problems  leading  to  such 
equations.)    (&)  Plane  geometry. 

Advanced  Studies  in  Mathematics — 

6.  Mathematics.— (a)  Logarithms;  plane  trigonometry,  with  its  applications  to  sur- 
veying and  navigation.  (&)  Either  solid  geometry  or  the  elements  of  analytic  geom- 
etry. 

7.  Mathemaiics.—{a)  Either  the  elements  of  analytic  geometry  or  solid  geometry. 
(h)  Either  elementary  mechanics  or  advanced  algebra. 

The  following  books  will  serve  to  indicate  the  nature  and  amount  of  the  require- 
ments in  logarithms  and  trigonometry,  analytic  geometry,  and  mechanics  : 

Logarithms  and  Trigonometry.  Wheelers  Logarithms  (Cambridge,  Sever)  or  the 
unbracketed  portions  of  Peirce's  Elements  of  Logarithms  (Boston,  Ginn  &  Co.). 


INFLUX  OF  FRENCH  MATHEMATICS.  149 

Wheeler's  Plane  Trigonometry  (same  publishers).  Problems  in  Plane  Trigonometry 
(Cambridge,  Sever).     Peirce's  Mathematical  Tables  (Boston,  Ginn  &  Co.). 

Analytic  Geometry.     Briggs's  Analytic  Geometry  (New  York,  Wiley  &  Co.). 

Mechanics.  Goodwin's  Elementary  Statics  (London,  Bell  &  Sons;  Cambridge, 
Sever). 

Advanced  Algebra.  Wentworth's  College  Algebra  (Boston,  Ginn  &  Co.),  to  arti- 
cle 498,  omitting  Chapters  XIX,  XX,  XXIY,  XXV,  XXVII,  XXVIII.  The  exami- 
nation will  be  mainly  occupied  with  the  portions  of  algebra,  as  thus  defined,  which 
are  not  included  in  the  elementary  requirement  in  algebra ;  but  elementary  questions 
are  not  necessarily  excluded. 

All  ia  all,  there  are  nine  "advanced"  studies  to  choose  from.  Since 
one  can  enter  the  college  after  passing  an  esamination  on  all  the  "ele- 
mentary" studies,  and  on  at  least  two  of  the  "advanced"  studies,  it 
follows  that  the  least  amount  of  mathematics  required  for  admission,  as 
a  regular  student,  is  that  stated  above  under  the  heading  "Elementary 
Studies  in  Mathematics." 

The  following  are  the 

Courses  of  instruction  in  mathematics. 
(1888-89.) 

A.  Logarithms. — Plane  Trigonometry,  with  its  applications  to  Surveying  and  Navi- 

gation.    Half-course.     Tu.,  Th.,  Sat,,  at  11  {first  half-year).    Professor  C.  J. 
White. 

B.  Analytic  Geometry  (elementary  course).    Half-course.    Ta.,  Th.,  Sat.,  at  11  {second 

half-year).    Professor  C.  J.  White. 

C.  Analytic  Geometry  (extended  course).  Man.,  Wed.,  Fri.,  at2.    Professor  Byerly. 

D.  Algebra.  Half-course.  Mon.,  Wed.,  FH.,  at  11  and  3  {firsthalf-year).  Mr.  Sawix. 
G.  Algebra  (extended  course).'  Half-course.    Tu.,  Th.,  Sat.,  at  10  {first  half-year).   Mr. 

Sawin. 
^.  Solid  Geometry.    Half-course.   Tu.,  Th.,  Sat.,  at  10  {secondhalf-year).    Mr.  Sawin. 
F.  Elementary  Mechanics.    Half-course.     Mon.,  Wed.,  Fri.,  at  12  {second  half-year), 
Mr.  Sawin. 
Not  to  be  given  after  1888-89. 

Courses  A,  B,  £^,and  F  correspond  to  Advanced  Mathematical  Studies  embraced, 
as  optional  studies,  in  the  examination  for  admission  to  college. 

1.  Practical  Applications  of  Plane  Trigonometry. — Spherical  Trigonometry. ^ — Appli- 

cations of  Spherical  Trigonometry  to  Astronomy  and  Navigation.    Wed.,  Fri., 
at  3.     Professor  C.  J.  White. 
Course  1  is  open  to  Freshmen  who  have  passed  the  examination  in  Plane  Trig- 
onometry. 

2.  Differential  and  Integral  Calculus  (First  Course).    Mon.,  Wed.,  Fri.,  at  11.    Pro- 

fessor C.  J.  White. 
Course  2  is  open  to  those  only  who  have  taken  Course  B  or  Course  C.  ■ 

3.  Analytic  Geometry ;  higher  course.    Mon.,   Wed.,  Fri.,  at  10.     Professor  J.  M. 

Peiece. 
Course  3  is  intended  for  students  who  have  taken  Course  C;  but  those  who 
have  taken  Course  B  may  elect  it,  if  deemed  qualified  by  the  instructor. 

4.  The  Elements  of  Mechanics.     Ta.,  Th.,  Sat.,  at  9.     Professor  B.  O.  Peirce. 

Course  4  is  intended  for  students  who  take  or  have  taken  Course  2. 

Candidates  for  Second- Year  Honors  may  take  Courses  2  and  3,  or  2  and  4. 
Other  courses  may  be  accepted  on  special  petition. 
5   Differential  and  Integral  Calculus  (Second  Course;.    Mon.,  Wed.,  Fri.,  at  11.   Pro- 
fessor Byerlt. 


150  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

[6.  Quaternions  and  Theoretical  Mechanics.    Mon.,  Wed.,  Fri.,  at  12,    Professor  J. 
M.  Peirce.] 
Omitted  in  1898-89. 
[7.  Higher  Plane  Curves.  ,  Professor  J.  M.  Peirce.] 

Omitted  in  1888-89. 
6.  Analytic  Mechanics.     Mon.,  Wed.,  Fri.,  at  10.    Professor  Byerly. 
9,  Quaternions  and  Theoretical  Mechanics  (Second  Course).    Mon.  at  12,  Fri.  11-1. 

Professor  J.  M.  Peirce. 
10.  Trigonometric  Series ;  Introduction  to  Spherical  Harmonics. — Theory  of  the  Po- 
tential.    Tu.,  Th.,  at  12,  Wed.  at  10.    Professors  Byerly  andB.  O.  Peirce. 
[11.  Hydromechanics.  Professor  B.  O.  Peirce.] 

Omitted  in  1888-89. 
13.  The  Theory  of  Functions.     Mo7i.  at  11,  Wed.  11-1.    Professsr  J.  M.  Peirce. 
20.  Special  Advanced  Study  and  Eesearch. — The  work  of  the  following  courses  wiU 
consist  in  investigations  and  reading,  to  be  carried  on  by  the  students  in  the 
courses,  under  the  guidance  of  the  instructors.    Students  will  be  expected  to 
present  their  results  from  week  to  week  in  the  form  of  lectures  and  theses. 
(«)  Questions  in  the  Theory  of  Functions.     Wed.,  3.30-5.30.     Professor  J.  M. 

Peirce. 
(b)  Higher  Algebra  (First  Course).    Mr.  Sawin. 

Some  few  studies  in  the  college  course  are  prescribed,  but  all  mathe- 
matical studies  are  elective.  No  mathematics  need  therefore  be  studied  in 
college.  A  student  can.  if  ho  chooses,  get  the  degree  of  bachelor  of 
arts  without  having  had  more  mathematics  than  plane  geometry  and 
algebra  through  quadratic  equations — the  minimum  requirement  for 
admission. 

We  conclude  this  article  by  quotations  from  a  letter  by  Prof.  L.  M. 
Hoskins,  of  the  University  of  Wisconsin,  who,  in  the  year  1884-85,  was 
honored  with  a  fellowship  at  Harvard,  and  studied  higher  mathematics 
there. 

"  There  were  two  courses,  in  '  quaternions  and  theoretical  mechanics,' 
given  by  Prof.  J.  M.  Peirce,  each  three  lectures  weekly  for  the  year. 
The  first  course  gave  the  elements  of  quaternions  and  the  dynamics  of 
a  particle,  covering  about  the  ground  of  Tait  and  Steele's  dynamics  of 
a  particle,  but  treated  by  quaternion  methods  largely.  The  second 
course  continued  with  higher  applications.  *  *  *  A  third  course,  on 
analytic  mechanics,  was  offered  by  Prof.  J.  jM.  Peirce,  consisting  of  lect- 
ures, following  BcDJamin  Peirce's  Analytic  Mechanics,  for  the  first  half- 
year,  and  for  the  second  Eouth's  Eigid  Dynamics,  the  part  relating  to 
moving  axes  and  relative  motion,  oscillations  about  equilibrium,  oscil- 
lations about  a  given  state  of  motion,  motion  of  a  rigid  body  under  any 
forces — in  short,  the  first  five  chapters  of  Volume  II.  *  *  *  I  at- 
tended also  a  course  in  "arbitrary  functions,"  by  Prof.  W.  E.  Byerly. 
This  covered  most  of  the  ground  of  Riemann's  Partielle  Differential' 
gleichungen.  The  main  subject  treated  is  the  methods  of  solution  of 
partial  differential  equations  subject  to  given  conditions,  a  class  of  prob- 
lems constantly  arising  in  physics.  The  course  naturally  includes  the 
proof  and  discussion  of  Fourier's  Theorem,  and  the  treatment  of  the  dif- 
ferent kinds  of  spherical  harmonics,  since  these  are  of  great  use  in  the 


INFLUX  OF  FEENCH  MATHEMATICS.  151 

solution  of  certain  classes  of  partial  differential  equations.  On  the  whole, 
I  found  this  as  attractive  a  part  of  pure  mathematics  as  I  ever  en- 
tered.   *    *    * 

"I  may  remark  that  the  branches  of  higher  mathematics  to  which 
most  attention  is  paid  at  Harvard  seem  to  be  theoretical  mechanics  and 
quaternions.  This  is  doubtless  due  to  the  influence  of  Benjamin  Peirce, 
whose  attainments  in  the  former  line  are  well  known,  and  who  was  also 
among  the  first  to  recognize  the  high  value  of  quaternion  methods. 
*  *  *  I  am  able  to  give  both  of  them  (Professors  J.  M.  Peirce  and 
Byerly)  high  praise  as  teachers  of  mathematics.  Both  are  clear,  logical 
lecturers,  and  popular  with  the  students.  *  *  *  Of  him  [Prof.  B.O. 
Peirce]  I  have  little  personal  knowledge,  but  am  sure  no  professor  was 
held  in  higher  estimation,  both  as  to  attainments  and  ability  as  a 
teacher.  *  *  *  In  1884-85  the  number  of  graduate  students  taking 
mathematics  was  five.    *    *    * 

'•A  mathematical '  seminar'  was  maintained  with  a  good  deal  of  in- 
terest, with  weekly  meetings  throughout  the  greater  part  of  the  year. 
These  meetings  were  under  the  supervision  of  the  mathematical  faculty, 
and  were  rather  informal  in  nature,  though  a  formal  programme  was 
usually  carried  out,  usually  by  volunteer  lectures  or  solutions." 

TALE  COLLEGE. 

The  successor  in  the  chair  of  mathematics  and  natural  philosophy  to 
the  lamented  Professor  Fisher  was  Matthew  E.  Button.  He  was  a 
graduate  of  the  college  and  was  professor  from  1822  to  1825.  He 
was  the  author  of  a  work  on  conic  sections  and  spherical  trigonometry. 
This  work  was  subsequently  revised  by  D wight,  who  "laid  the  students 
and  teachers  of  that  day  under  everlasting  obligations  by  his  simplifi- 
cation and  abbreviation  of  those  endless  algebraic  formulse  in  Button's 
Conic  Sections." 

From  1825  to  1836  Benison  Olmsted  occupied  the  chair  which  had 
been  made  vacant  by  the  death  of  Professor  Button.  He  was  born  in 
Hartford,  Conn.,  in  1791,  graduated  at  Yale  in  1813,  became  tutor  there 
in  1815,  was  elected  professor  of  chemistry  at  the  University  of  E'orth 
Carolina  in  1815,  and  finally  returned  thence  to  assume  the  duties  of 
the  chair  of  mathematics  and  physics  at  Yale.  He  had  made  natural 
philosophy  and  chemistry  his  specialty,  and  possessed  no  special  fitness 
for  the  teaching  of  mathematics. 

Professor  Olmsted  was  renowned  as  a  teacher  rather  than  an  orig- 
inal investigator.  His  teaching  power  was  indeed  great,  and  he  exerted 
a  beneficial  influence,  not  only  in  college,  but  also  upon  the  education 
in  the  common  schools  of  Connecticut.  In  1831  appeared  his  Natural 
Philosophy,  which  superseded  the  antiquated  work  of  Enfield.  In  the 
next  year  was  written  his  School  Philosophy,  a  more  elementary  work  j 
and  in  1839  his  Astronomy.  He  wrote  also  the  Eudimeuts  of  !N"atural 
Philosophy  and  Astronomy,  which  passed  through  some  fifty  editions. 


152  TEACHING   AND   HISTORY    OP   MATHEMATICS. 

His  ]S'atural  Philosophy  and  Astronomy  came  to  be  almost  universally 
used  in  our  colleges.  The  Philosophy  was  later  revised  by  Prof.  E.  S. 
Snell,  of  Amherst  College,  and,  still  later,  by  Prof.  J.  Fickliu,  of  the 
University  of  Missouri.  In  point  of  scientific  accuracy  Olmsted's  works 
were  sometimes  rather  defective.   They  were  somewhat  old-fashioned. 

As  early  as  1830  a  good  telescope  of  moderate  size  was  procured  at 
Yale  College.  For  want  of  an  observatory  it  was  difficult  to  make  ac- 
curate observations  with  it.  But  it  served,  nevertheless,  the  excellent 
purpose  offurnishing  a  means  of  observing  the  great  November  shower 
of  meteors,  which  occurred  not  long  afterward.  These  showers,  Hal- 
ley's  Comet,  and  the  telescope  enabled  Professor  Olmsted  to  arouse  a 
great  deal  of  astronooiical  enthusiasm  at  Yale,  and  for  a  few  years  a 
number  of  students  turned  their  attention  to  astronomy.  Of  the  math- 
ematicians and  astronomers  who  graduated  in  those  days  are  Stanley 
and  Mason,  long  since  deceasetl ;  and  Loomis  and  Lyman,  who  are  now 
aged  professors  at  Yale.  The  ablest  mathematician  and  astronomer 
which  Yale  has  produced  is  William  Chauvenet.  As  a  teacher  he  was 
never  connected  with  his  alma  mater^  though  a  professorship  was  of* 
fered  him  twice. 

In  1835  the  chair  of  mathematics  and  natural  philosophy  was  divided 
into  two  separate  ones — Olmsted  retaining  that  of  natural  philosophy, 
and  Anthony  D.  Stanley  being  elected  to  that  of  mathematics. 

One  of  the  most  prominent  of  early  tutors  in  mathematics  at  Yale  was 
F.  A.  P.  Barnard,  of  the  class  of  1828,  which  was  known  as  a  "mathe- 
matical class,"  for  the  mathematical  talent  it  embraced.  While  tutor  at 
Yale  he  prepared  an  edition  of  Bridge's  Conic  Sections,  in  which  the 
work  was  substantially  rewritten  and  also  considerably  enlarged. 

We  proceed  to  examine  the  mathematical  courses  during  the  time 
of  Professors  Dutton  and  Olmsted. 

In  1824,  Arithmetic  was  the  requirement  in  mathematics  for  admis- 
sion ;  in  1833,  "  Barnard's  or  Adam's  Arithmetic  j "  in  1845,  Arithmetic 
and  Day's  Algebra  to  quadratics. 

The  mathematical  course  for  1824  was,  for  Freshmen,  Day's  Algebra 
during  the  first  two  terms,  with  no  mathematics  for  the  third  term  j  for 
Sophomores,  six  books  of  Playfair's  Euclid  during  the  first  and  part  of 
the  second  term,  and  Day's  Mathematics  (including  plane  trigonometry, 
logarithms,  mensuration  of  surfaces  and  solids,  isoperimetry,  navigation 
and  surveying)  and  Dutton's  Conic  Sections,  during  the  rest  of  the  year  j 
for  Juniors,  Dutton's  Spherical  Trigonometry  during  the  first  term,  and 
Enfield's  Astronomy  and  Vince's  Fluxions  during  the  third  term.  The 
Seniors  had  no  mathematics  in  their  course.  In  1825  the  study  of  Euclid 
was  begun  at  the  close  of  the  Freshman  year.  Vince's  Fluxions  still 
appeared  in  that  year  as  a  text-book. 

The  writer  has  not  been  able  to  see  catalogues  for  the  years  1826-32. 
In  1833,  Olmsted's  Natural  Philosophy  was  in  use.  '^Fluxions"  were 
also  named,  but  this  meant  then,  most  likely,  the  differential  and  in- 


INFLUX  OF  FRENCH  MATHEMATICS,  153 

tegral  calculus.  In  the  Sophomore  year,  Bridge's  Conic  Sections  (prob- 
ably Barnard's  edition)  was  used  in  place  of  Dattoa's.  No  changes  in 
the  course  were  made  for  several  years  after. 

The  teaching  of  mathematics  to  the  two  lower  classes  in  college  was 
generally  intrusted  to  young  and  inexperienced  tutors,  who  had,  as  a 
rule,  a  very  meagre  acquaintance  with  the  subjects  which  they  were 
supposed  to  teach.  It  is  therefore  not  surprising  that  poor  results 
were  generally  reached,  and  that  the  study  of  mathematics  was  very 
unpopular.  Especially  unpopular  was  the  study  of  conic  sections. 
IITo  eiforts  seem  to  have  been  made  on  the  part  of  tutors  to  make  this 
study  more  attractive  and  to  show  its  usefulness,  by  pointing  out  its 
application  in  the  study  of  physics  and  astronomy.  Moreover,  the  old 
books  on  conic  sections  were  as  dry  as  dust.  The  dissatisfaction  among 
students  finally  culminated,  in  1830,  in  what  is  known  as  the  "  conic 
sections  rebellion."  Eebellions  among  students  were  then  not  unfre- 
quent.  Some  years  previously  had  taken  place  the  "bread  and  butter 
rebellion,"  caused  by  the  poor  quality  of  board  that  the  students  were 
receiving.  Neither  their  physical  nor  their  intellectual  food  seems  to 
have  been  palatable  to  them.  The  "conic  sections  rebellion"  was  a 
refusal,  on  the  part  of  the  Sophomores,  to  recite  in  the  manner  pre- 
scribed by  the  college  rules.  They  petitioned  that  the  method  of  rec- 
itation required  by  the  college  be  changed,  that  they  might  "  explain 
conic  sections  from  the  book,  and  not  demonstrate  them  from  the 
figure."*  We  should  judge  from  this  that  the  practice  had  hitherto 
prevailed  of  simjjly  asking  the  student  to  explain  certain  parts  of  the 
subject,  with  the  book  open  before  him,  without  requiring  him  to  go  to 
the  blackboard  (if  blackboards  were  used)  to  explain  the  lesson  from 
his  own  figure  independently  of  the  book.  We  have  not  been  able  to 
ascertain  at  what  time  the  blackboard  was  introduced  into  the  mathe- 
matical class-room  at  Yale,  but  it  is  not  unlikely  that  the  above  rebel- 
lion arose  in  the  attempt,  on  the  part  of  the  faculty,  to  introduce  such 
improved  methods  as  the  use  of  the  blackboard  would  suggest.  The 
new  methods  may  have  called  for  greater  effort  on  the  part  of  the  stu- 
dent, and  may  thus  have  brought  about  the  "  rebellion." 

The  general  impression  prevailed  at  Yale,  in  those  days,  that  the 
mathematical  course  there  was  a  very  difficult  and  thorough  one.  "  This 
fancy  certainly  derived  some  support  from  comparison  with  the  class- 
ical course,  as  compared  with  ivhich  the  mathematical  was  undoubtedly 
a  good  one."f 

Mr.  Bristed,  who  entered  Yale  in  1835,  says  that  in  mathematics  the 
classes  studied  books  rather  than,  subjects,  and  crammed  from  one  day 
to  another.  "  A  great  deal  of  the  work,"  says  he  "  of  the  second  and 
third  year  consisted  of  long  calculations  of  examples  worked  witb  loga- 
rithms, which  consumed  a  great  deal  of  time  without  giving-  any  insight 

*  Yale  College :  a  Skerch  of  its  History,  by  William  L.  Kingsley,  p.  137. 

tFive  Years  in  English  Universities,  by  Charles  Astor  Bristed,  3d  ed.,  1873,  p.  456. 


154  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

into  principles,  and  were  equally  distasteful  to  the  good  and  the  bad 
matheinaticiaus."  "  They  (the  best  mathematicians)  complained  that 
with  the  exception  of  two  prizes  for  problems  during  the  Freshman  and 
Sophomore  years,  and  an  occasional  ^original  demonstration'  in  the 
recitation  room,  they  had  no  chance  of  showing  their  superior  ability 
and  acquirements;  that  much  of  their  time  was  lost  in  long  arithmetical 
and  logarithmical  computations;  that  classical  men  were  continually 
tempted  to  '  skin '  (copy)  the  solutions  of  these  examples,  and  thus  put 
themselves  unjustly  on  a  level  with  them."  The  bad  practice  of  giving 
long  and  tedious  examples  to  work  has  been  quite  prevalent  in  our 
colleges  until  within  recent  years,  especially  in  trigonometry.  For  or- 
dinary class- work  four-place  logarithmic  tables  are  sufficient,  we  should 
think.  Prof.  J.  M.  Peirce,  of  Harvard,  has  done  much  toward  inaugu- 
rating a  reform  in  this  matter,  by  his  publication  of  four-place  tables. 
Such  tables  are  of  sufficient  accuracy  even  in  connection  with  the  ordi- 
nary physical  experiments  which  the  student  may  make  in  the  labora- 
tory. 

In  1836  Anthony  D.  Stanley  became  professor  of  mathematics.  He 
held  this  place  until  his  death  in  1853.  He  was  a  native  of  Connecti- 
cut and  graduated  at  Tale  in  1830.  Two  years  later  he  was  appointed 
tutor  and  afterward  professor.  We  are  told  that  Professor  Stanley 
took  special  interest  in  the  theory  of  numbers,  and  that  he  had  once  an 
excellent  occasion  to  show  his  skill.  "  In  1835  an  anonymous  writer  in 
the  Stamford  Sentinel  challenged  the  entire  faculty  of  Tale  to  arrange 
the  nine  digits  in  such  order  that  their  square  root  could  be  extracted 
without  a  remainder.  In  a  few  days  Mr.  Stanley  over  the  signature 
"X"  gave  the  required  solution,  and  added  that  the  question  admitted 
of  more  than  one  answer,  and  called  upon  the  proposer  to  produce 
them.  To  this  challenge  his  opponent  made  an  evasive  reply,  in  which 
he  stated  the  number  of  solutions  to  be  nine,  but  did  not  communicate 
any  solution."  *  Stanley  found  twenty-eight  different  solutions,  but  even 
a  larger  number  is  possible. 

It  seems  that  when  Stanley  was  appointed  professor  he  did  not  im- 
mediately enter  upon  the  discharge  of  the  duties  of  the  chair,  but  went 
to  Europe  two  years  and  spent  most  of  the  time  in  Paris,  where  he  at- 
tended lectures  at  the  Sorbonne  and  College  of  France.  In  1846  he 
published  Tables  of  Logarithms,  which  were  uncommonly  accurate.  In 
1848  appeared  his  Elementary  Treatise  on  Spherical  Geometery  and 
Trigonometry.  In  the  preceding  year  he  published  an  article  in  the 
American  Journal  of  Science,  "  On  the  Variation  of  a  Differential  Co- 
efficient of  aFunction  of  any  Kumber  of  Variables."  "  In  this  memoir," 
says  Professor  Loomis,  "  he  resolves  a  problem  which  had  already  oc- 
cupied the  attention  of  La  Grange,  Poisson,  Ostrogradsky,  and  Pagani, 
the  latter  of  whom  was  the  only  one  who  obtained  a  correct  solution  of 
it.    Professor  Stanley  here  gives  a  solution  of  the  same  problem  more 

Yale  College :  a  Sketch  of  its  History,  by  William  L.  Kingsley, 


INFLUX  OP  FRENCH  MATHEMATICS.  155 

simple  and  concise  than  Pagani's,  and  which  was  discovered  before  re- 
ceiving the  solution  of  that  mathematician."*  In  1849  he  suliered  from 
a  severe  cold,  and  he  sought  relief  in  Italy  and  Egypt.  On  his  return 
he  assisted  in  completing  the  revision  of  Day's  Algebra,  which  he  had 
begun  before  leaving.     He  died  in  1853. 

Somewhat  later  than  Prof.  Charles  Davies,  Prof.  Elias  Loomis  began 
the  publication  of  mathematical  text-books,  which,  like  Davies'  works, 
became  extremely  popular  throughout  the  United  States.  Professor 
Loomis  has  been  connected  with  Yale  College  since  1860,  but  not  as 
professor  of  mathematics.  Indeed,  his  specialty  has  not  been  mathe- 
matics. His  original  contributions  to  science  have  been  in  other  fields. 
At  Yale  he  has  been  professor  of  natural  philosophy  and  astronomy. 
His  chief  scientific  work  has  been  as  a  meteorologist  and  astronomical 
observer.  In  his  younger  days  Professor  Loomis  was  a  man  of  wonder- 
ful activity,  but  now  he  is  nearly  four  score  years  old  and  an  invalid. 

Professor  Loomis  was  born  in  Connecticut  in  1811 ;  was  graduated  at 
Yale  College  in  1830.  Af.er  graduating  he  occasionally  contributed 
solutions  to  Eyan's  Mathematical  Diary.  He  was  for  a  time  tutor  in  Yale. 
Together  with  Professor  Twining,  of  West  Point,  he  made  observations 
for  determining  the  altitude  of  shooting-stars.  These  were,  most  likely, 
the  first  concerted  observations  of  the  kind  made  in  America.  He  was 
the  first  one  in  America  to  discover  Halley's  Comet  in  1835.  The  next 
year  he  was  chosen  professor  of  mathematics  and  natural  philosophy  at 
the  Western  Eeserve  College,  with  permission  to  spend  a  year  in  Eu- 
rope. In  Paris  he  attended  lectures  of  Arago,  Biot,  Poisson,  Dulong, 
Pouillet,  and  others.  He  returned  with  astronomical,  physical,  and 
meteorological  instruments,  and  during  the  next  season  an  astronomical 
observatory  was  erected  at  the  Western  Eeserve  College,  in  Ohio.  Only 
three  observatories  existed  in  the  United  States  before  this,  namely,  at 
the  University  of  North  Carolina,  at  Yale,  and  at  Williams  College. 

In  1844  Professor  Loomis  became  professor  in  the  University  of  the 
City  of  New  York.  "Having  here  no  instruments  for  observation,  he 
was  induced  to  undertake  the  preparation  of  a  text- book  on  algebra  j 
especially  designed  for  the  use  of  his  own  classes.  This  book  prepared 
the  way  for  a  second,  and  the  second  was  followed  by  a  third,  until,  ul- 
timately, his  text-books  embraced  the  whole  range  of  mathematics  and 
natural  philosophy,  astronomy,  and  meteorology."*  His  principal 
mathematical  and  astronomical  text-books  are,  Plane  and  Spherical 
Trigonometry,  1848;  Analytical  Geometry  and  Calculus,  1857;  Ele- 
ments of  Algebra,  1851 ;  Elements  of  Geometry  and  Conic  Sections, 
1851 ;  Practical  Astronomy,  1855 ;  Elements  of  Arithmetic,  1863.  His 
treatise  on  astronomy,  now  obsolescent,  received  in  its  day  high  com- 
mendation from  leading  astronomers.  Some  of  his  mathematical  text- 
books were,  at  first,  very  thin,  but  were  gradually  enlarged  in  subse- 

*  Yale  College ;  a  Sketcli  of  its  History,  by  William  L.  Kingsley. 


156  TEACHING   AND   HISTORY   OF   MATHEMATICS 

quent  editions.  Thus,  his  Analytical  Geometry  and  Calculus  were  at 
first  combined  in  one  small  volume,  v/hile,  subsequently,  the  two  sub- 
jects were  published  separately  in  volumes  about  as  large,  each,  as  the 
earlier  combined  volume. 

The  books  of  Loorais  were  written  in  a  clear,  simple  style,  and  were 
well  adapted  for  use  in  the  class-room.  There  was  nothing  in  them 
which  any  student  of  ordinary  ability  and  application  did  not  readily 
master.  These  characteristics  made  Loomis's  works  very  popular.  A 
student  desiring  to  secure  a  somewhat  extended  knowledge  of  the  vari- 
ous mathematical  subjects  would  hardly  have  found  Loomis's  works 
to  answer  his  i^urpose ;  nor  would  the  works  of  Davies  have  given  him 
better  satisfaction.  He  would  have  found  more  of  what  he  wanted  in 
the  books  of  Peirce  and  Chauvenet.  Nor  were  Loomis's  works  always 
up  with  the  times.  The  treatment  of  series  is  bad,  both  in  Ms  Algebra 
and  in  his  Calculus.  Again,  take  the  following  statement,  for  instance: 
"  ]S'o  general  solution  of  an  equation  higher  than  the  fourth  degree  has 
yet  been  discovered."*  This  piece  of  historical  information  is  unsatis- 
factory ;  for,  in  the  first  place,  M.  Hermite  has  given  a  transcendental 
solution  of  the  quintic  and,  in  the  second  place,  Abel  and  Wantzel  have 
proved  that  an  algebraic  solution  of  equations  higher  than  the  fourth 
degree  is  impossible.  Perhaps  the  best  mathematical  work,  in  point  of 
accuracy,  is  his  Elementary  Geometry.  It  has  been  said  of  American 
writers  that,  while  they  have  given  up  Euclid,  they  have  modified  Le- 
gendre's  Geometry  so  as  to  make  it  resemble  Euclid  as  much  as  possi- 
ble. This  applies  to  Loomis  with  greater  force,  perhaps,  than  to  any 
other  author.  His  trigonometry  has  been  wedded  to  the  old  "  line  sys- 
tem," and  it  is  only  within  the  last  two  or  three  years  that  a  divorce 
has  been  secured. 

TVhile  Loomis  has  made  no  original  contributions  to  pure  mathemat- 
ics, he  has  not  been  idle  in  other  lines  of  research.  He  has  contributed 
one  hundred  or  more  papers  (chiefly  on  astronomical,  meteorological, 
and  physical  subjects)  to  the  American  Philosophical  Society,  Connecti- 
cut Academy,  Smithsonian  Institution,  American  Journal  of  Science, 
and  Gould's  Astronomical  Journal.  Some  of  his  papers  have  been  re- 
printed in  Europe.  His  Contributions  to  Meteorology  was  translated 
into  French. 

Professor  Stanley's  successor  in  the  mathematical  chair  at  Yale  is 
Professor  Hubert  Anson  Newton.  He  graduated  at  Tale  in  1850,  after 
which  he  studied  higher  mathematics.  In  1852  he  was  made  tutor,  and 
when  he  entered  upon  that  office  in  1853  he  was  given  charge  of  the 
entire  mathematical  department  at  once,  owing  to  the  illness  of  Profes- 
sor Stanley.  In  1855  he  was  elected  full  professor,  with  permission  to 
spend  one  year  abroad.  In  1856  he  began  the  active  discharge  of  the 
duties  of  the  chair,  which  he  still  holds.  Professor  Newton's  publica- 
tions have  been  restricted  almost  exclusively  to  scientific  papers,  which 

*  Treatise  on  Algebra,  1873,  p.  334. 


INFLUX  OF  FRENCH  MATHEMATICS.  J  57 

have  appeared  in  the  Memoirs  of  the  National  Academy  of  Sciences  and 
in  the  American  Journal  of  Science.  He  is  best  known  to  science  for 
his  observations  on  shooting-stars  and  star-showers.  He  wrote  for  the 
Encyclopaedia  Britannica  the  article  on  "  Meteorites."  His  work  iu 
pure  mathematics  includes  a  paper  "  On  the  Construction  of  Certain 
Curves  by  Points,"  published  in  the  Mathematical  Monthly,  and  on 
"  Certain  Transcendental  Curves." 

Since  1871  Eugene  L.  Eichards  has  been  assistant  professorof  mathe- 
matics.   He  is  the  author  of  a  Trigonometry. 

In  1873  John  E.  Clark,  who  had  been  professor  at  the  University  of 
Michigan,  was  chosen  professor  of  mathematics  at  Tale.  Since  1881 
Andrew  W.  Phillips  has  been  assistant  professor  of  mathematics ;  also 
William  Beebe  since  1882.  Phillips  and  Beebe  have  written  a  novel 
and  successful  treatise  on  Graphic  Algebra.  i 

In  1871  J.  Willard  Gibbs  was  elected  professor  of  mathematical 
physics.  He  graduated  at  Yale  in  1858,  and  after  graduation  contin- 
ued his  mathematical  and  physical  studies.  He  was  tutor  from  1863 
to  1866.  Afterward  he  went  to  Europe  and  spent  three  years  in  study 
at  Paris,  Berlin,  aud  Heidelberg.  Much  of  his  time  has  been  given  to 
thermodynamics.  He  contributed  in  1873  to  the  Connecticut  Academy 
an  article  on  Graphical  Methods  in  Thermodynamics  of  Fluids.  In  the 
same  year  appeared  A  Method  of  Geometric  Eepresentation  of  the  Ther- 
modynamic Properties  of  Substances  by  Means  of  Surfaces. 

But  Professor  Gibbs's  studies  have  been  carried  on  also  in  the  field  of 
pure  mathematics.  He  has  published  a  treatise  on  the  Elements  of 
Vector  Analysis,  which  is  a  triple  algebra,  as  distioguished  from  quater- 
nions, a  quadruple  algebra.  Vector  analysis  has  been  applied  by  Pro- 
fessor Gibbs  to  about  the  same  kiud  of  problems  as  quaternions.  The 
advantage  claimed  for  vector  analysis  over  quaternions  is  that  the 
former  reaches  solutions  more  simply  and  directly,  and  that  its  prin- 
ciples can  be  developed  more  concisely.  In  1886  Professor  Gibbs  read 
an  exceedingly  interesting  paper  before  the  American  Association  for 
the  Advancement  of  Science  on  Multiple  Algebra,  which  contains  an 
excellent  sketch  of  the  development  of  this  science  in  the  hands  of  Grass- 
man,  Hamiltou,  Hankel,  Beujamin  Peirce,  Sylvester,  Cayley,  and  others. 
As  to  the  applications  of  multiple  algebra,  Professor  Gibbs  says:* 

"Maxwell's  Treatise  on  Electricity  and  Magnetism  has  done  so  much 
to  familiarize  students  of  physics  with  quaternion  notations  that  it 
seems  impossible  that  this  subject  should  ever  again  be  entirely  divorced 
from  the  methods  of  multiple  algebra. 

"  I  wish  that  I  could  say  as  much  of  astronomy.  It  is,  I  think,  to  be  re- 
gretted that  the  oldest  of  the  scientific  applications  of  mathematics,  the 
most  dignified,  the  most  conservative,  should  keep  so  far  aloof  from  the 
youngest  of  mathematical  methods." 

We  now  return  to  the  courses  of  study  at  Yale  College.    The  catalogue 

*  Proceedings  American  Association  for  the  Advancement  of  Science,  1886,  p.  62. 


158  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

of  1845  shows  that  "Day's  Algebra  to  quadratics"  was  added  to 
"  arithmetic  "  as  a  requirement  for  admission  to  college.  In  1852  Thom- 
son's was  the  arithmetic  recommended.  In  1855  the  requirements  were 
again  increased  by  the  addition  of  two  books  in  Playfair's  Euclid.  In 
1870  the  terms  were  higher  arithmetic,  Loomis's  Algebra  to  quadratics, 
and  two  books  of  Playfair's  Euclid  (or  the  first,  third,  and  fourth 
books  of  Davies'  Legendre,  or  of  Loomis'  Elements  of  Geometry) ;  in 
1885,  arithmetic,  algebra  as  far  as  logarithms  in  Loomis,  first  book  in 
Euclid,  and  the  first  thirty-three  exercises  thereon  in  Todhunter's  edition 
(or  the  first  four  books  in  another  geometry) ;  in  1887,  higher  arithmetic 
(including  the  metric  system  of  weights  and  measures),  algebra  (Loomis 
as  far  as  logarithms),  plane  geometry.  All  candidates  for  admission 
are  examined  on  the  same  studies,  no  matter  what  courses  they  may 
wish  to  pursue  in  college.  It  is  also  worthy  of  remark  that,  since  1885, 
the  use  of  Euclid  as  a  text-book  in  geometry  has  been  discontinued  at 
Yale,  and  Princeton  is  now  the  only  prominent  college  in  the  country 
which  still  adheres  to  Euclid. 

We  come  now  to  the  mathematical  course  in  college.  In  1848  it  was 
as  follows : 

Freshmen,  Day's  Algebra,  Playfair's  Euclid;  Sophomores,  Day's  Math- 
ematics, Bridge's  Conic  Sections,  and  Stanley's  Spherical  Geometry 
and  Trigonometry;  Juniors,  Olmsted's  Natural  Philosophy,  Mechanics, 
Hydraulics,  Hydrostatics,  Olmsted's  Astronomy,  Analytical  Geometry 
or  Fluxions  (optional). 

Fluxions  seem  to  have  been  optional  all  the  time,  though  in  previous 
catalogues  they  appear  as  a  regular  study.  Analytical  geometry  was 
also  optional  for  that  year.  In  1852  Loomis's  Analytical  Geometry 
and  Calculus  appear  in  the  catalogue  as  Sophomore  studies.  In  1854 
Bridge's  Conic  Sections  or  Analytical  Geometry  appear  as  part  of  the 
work  of  the  Sophomore  year ;  and  Church's  Difierential  Calculus  in  the 
Junior  year.  But  analytical  geometry  and  calculus  were  elective  stud- 
ies. "  Those  desirous  of  pursuing  higher  mathematics  are  allowed  to 
choose  analytical  geometry  in  place  of  regular  mathematics  in  the  third 
term  Sophomore,  and  calculus  in  the  Junior  for  Greek  and  Latin," 

In  1858  Loomis's  Calculus  is  given  in  the  Sophomore  year,  and 
Todhunter's  in  the  Junior. 

The  course  was  as  follows  in  1870 :  Freshmen,  Loomis's  Algebra,  Play- 
fair's Euclid,  Loomis's  Conic  Sections;  Sophomores,  Loomis's  Trigonom- 
etry, Stanley's  Spherical  Geometry,  Davies'  Analytical  Geometry; 
Juniors,  Calculus,  Loomis's  Astronomy.  The  next  year,  Chauvenet's 
Geometry  was  used  with  Euclid  in  the  Freshman  class. 

In  1885  the  course  was — Freshmen,  Todhunter's  Euclid  (Books  III 
and  IV),  Chauvenet's  Geometry,  Eichard's  Plane  Trigonometry,  Phillips 
and  Beebe's  Graphic  Algebra;  Sophomores,  Loomis's  Analytical  Ge- 
ometry (plane  and  solid),  Dana's  Mechanics;  Juniors,  Loomis's  Astron- 
omy (required).  Calculus,  Geodesy,  Descriptive  Geometry  (all  three 
elective)  j  Seniors,  Calculus,  Vector  Analysis  (both  elective). 


INFLUX  OF  FRENCH  MATHEMATICS.  159 

The  course  for  the  year  1887-88  is  substantially  the  same  as  that  of 
1885.    We  quote  from  the  catalogue  the  following  account  of  it : 

"In  geometry  the  exercises  consist  in  recitations  from  the  text-book, 
the  original  demonstration  of  theorems,  and  applications  of  the  prin- 
ciples to  the  solution  of  numerical  problems. 

"After  the  student  has  gained  facility  in  the  use  of  trigonometrical 
tables,  the  principles  of  plane  trigonometry  are  applied  to  the  problems 
of  mensuration,  surveying,  and  navigation,  and  those  of  spherical  trigo- 
nometry to  the  elementary  problems  relating  to  the  celestial  sphere. 

"In  algebra  the  elementary  principles  of  the  theory  of  equations 
are  illustrated  graphically,  and  the  student  is  exercised  in  the  numerical 
solution  of  equations  of  the  higher  degrees  and  the  graphical  represen- 
tation of  the  relations  of  quantities. 

"  In  analytical  geometry  the  student  is  carried  through  the  elementary 
properties  of  the  lines  and  surfaces  of  the  second  degree,  and  is  intro- 
duced to  the  theory  of  map  projection. 

"These  are  studies  of  the  Freshman  and  Sophomore  years,  and,  to- 
gether with  the  elements  of  astronomy  which  are  pursued  in  Junior  year, 
are  regarded  as  essential  parts  of  a  liberal  education. 

"In  the  Junior  and  Senior  years  opportunity  is  given  in  the  elective 
courses  to  obtain  a  wider  knowledge  of  analytical  geometry  and  trigo- 
nometry with  their  applications  to  geodesy  and  astronomy.  A  longer 
and  shorter  course  are  provided  in  Junior  year  in  differential  and  integral 
calculus.  The  shorter  course  is  designed  for  students  who  desire  to  be- 
come acquainted  with  the  methods  of  the  calculus,  but  whose  principal 
studies  are  not  of  a  mathematical  character.  The  longer  course  is  de- 
signed for  such  as  expect  to  make  a  serious  study  of  any  department  of 
pure  or  applied  mathematics. 

"  In  Senior  year  advanced  subjects  in  the  calculus  and  the  elements 
of  analytical  mechanics  form  one  line  of  study. 

"  An  elementary  and  an  advanced  course  are  provided  in  what  is 
called  vector  analysis.  The  object  of  these  courses  is  to  introduce  the 
student  to  the  methods  of  multiple  algebra  in  geometry,  mechanics, 
and  physics.  The  matter  taught  is  not  entirely  unlike  that  usually 
given  in  courses  in  quaternions,  but  the  method  followed  is  in  some  re- 
spects nearer  to  Grassmann's  than  to  Hamilton's.  The  elementary 
course  is  confined  to  the  simplest  algebraic  relations  of  vectors.  The 
advanced  course  includes  differentiation  with  respect  to  position  in 
space,  and  the  theory  of  linear  vector  functions. 

"Students  who  show  special  aptitude  are  exercised  in  the  working  up 
of  subjects  which  require  the  use  of  the  library  and  more  prolonged 
investigation  than  the  daily  exercises  of  the  class-room.  Such  work 
begins  in  Freshman  year.  There  is  a  considerable  collection  of  models, 
which  are  used  to  assist  the  imagination  in  the  various  branches  of 
study." 

In  November,  1877,  a  Mathematical  Club  was  formed  at  Yale.    Pro- 


160  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

fessor  Gibbs  has  been  the  leading  spirit  in  it.  He  has,  in  recent  years, 
presented  papers  showing  the  application  of  vector  analysis  to  the  com- 
putation of  elliptic  orbits.  The  work  of  the  club  has,  however,  not  been 
confined  to  pure  mathematics.  Professor  Newton  has  presented  sub- 
jects ou  meteors  and  astronomy,  and  Professor  Hastings  has  given  re- 
sults of  experiments  made  by  him  on  light. 

COLLEGE  OF  NEW  JERSEY. 

In  1830  Albert  B.  Dod  became  professor  of  mathematics.  He  seems 
to  have  been  a  favorite  teacher.  His  pupils  cherish  fondly  the  recollec- 
tion of  "his  brilliant  genius  and  the  interest  which  he  infused  into  the 
study  of  higher  mathematics,  as  well  as  the  magnetic  charm  of  his  man- 
ner, as  by  the  wonderful  acuteness  and  perspicuity  with  which  he  mas- 
tered and  explained  the  most  abstruse  problems."  The  same  qualities 
shone  attractively  in  his  lectures  on  architecture.*  He  discharged  the 
duties  of  his  office  with  signal  ability  till  his  death,  in  1845.  The  family 
to  which  he  belonged  had  for  several  generations  been  remarkable  both 
for  mathematical  taste  and  talent.  His  father  constructed  the  engine 
of  the  Savannah,  the  first  steam-boat  that  crossed  the  Atlantic. 

The  scientific  and  mathematical  departments  of  Princeton  were  first 
made  prominent  by  the  labors  of  Professors  Henry  and  Alexander. 
Stephen  Alexander  was  graduated  at  Union  College,  New  York,  in 
1824,  a^t  the  age  of  eighteen,  v/ith  high  honors.  He  then  engaged  in 
teaching.  In  1830  and  1831  he  was  in  Albany  making  numerous  astro- 
nomical observations  and  communicating  them  to  the  Albany  Institute. 
He  and  Joseph  Henry  were  relatives.  "  Professor  Henry  was  a  son 
of  the  elder  Alexander's  sister,  and  in  1830  he  married  his  cousin,  Miss 
Alexander,  thus  establishing  a  double  relationship,  which  unquestion- 
ably shaped  the  whole  life  and  fortune  of  his  younger  and  favorite 
cousin  and  brother-in-law.t  In  1832  Professor  Henry  was  elected  to  the 
chair  of  natural  philosophy  at  the  College  of  New  Jersey.  Alexander 
went  with  Henry  and  his  family  to  Princeton.  He  there  entered  the 
Theological  Seminary  as  a  student,  but  in  1833  he  was  appointed  tutor 
in  the  college.  "  In  1834  he  was  made  adjunct  professor  of  mathematics, 
and  in  1840  he  was  promoted  to  the  full  professorship  of  astronomy, 
which  he  retained  until  187G.  During  the  long  intervening  period  the 
style  and  duties  of  his  professorshij)  were  several  times  more  or  less 
modified.  For  several  years  after  the  death  of  Professor  Dod  he  was 
professor  of  mathematics  and  astronomy.  When  Professor  Henry  went 
to  Washington  he  gave  up  the  mathematics  and  became  professor  of  natu- 
ral philosophy  and  astronomy,  but  he  always  held  fast  to  astronomy." 

In  1847  John  Thomas  Duffield  became  connected  with  the  mathemati- 

*  The  Princeton  Book,  1879. 

t  Biographical  Memoirs  of  the  National  Academy  of  Sciences,  Vol.  II,  p.  886,  "  Bio- 
graphical Memoir  of  Stephen  Alexander,"  by  C.  A.  Young.  Our  remarks  on  Professor 
Alexander  are  drawn  chiefly  from  this  sketch. 


INFLUX  OF  FEENCH  MATHEMATICS.  161 

cal  department.  He  graduated  at  Princeton  College  in  1841,  afterward 
studied  theology,  and  then  was  appointed  tutor  in  Greek.  From  1847 
to  1850  he  served  as  adjunct  professor  of  mathematics.  During  two 
years  he  had  charge  of  a  Presbyterian  church  in  connection  with  his 
duties  in  the  college.  He  published,  also,  a  volume  of  sermons.  He 
has  been  professor  of  mathematics  since  1850.  For  many  years  the 
mathematical  teaching  at  Princeton  was  in  the  hands  of  Professor  Duf- 
field  and  Professor  Alexander.  The  former  possessed  great  power  in 
teaching  young  students,  while  the  latter  led  their  way  into  the  more 
advanced  mathematics  and  astronomy. 

In  1850  the  requirements  for  admission  were  arithmetic  and  the  ele- 
ments of  algebra  through  simple  equations.  The  Freshmen  studied 
Hackley's  Algebra  and  Playfair's  Euclid;  the  SopJiomores  finished  Euclid 
and  then  took  up  plane  and  spherical  trigonometry,  navigation,  etc. ; 
the  Juniors  studied  analytical  geometry  (Young's),  Alexander's  Differ- 
ential and  Integral  Calculus,  and  mechanics. 

Princeton  is  one  of  the  very  few  colleges  in  this  cbuntry  which  have 
retained  Euclid  as  a  text-book  in  geometry  to  the  present  day.  "  Euclid 
is  used  as  a  text-book  in  geometry  because  of  its  historical  associations 
and  its  decided  superiority  for  the  purpose  of  mental  discipline  to  any 
modern  textbook."* 

Eev.  Dr.  E.  G.  Hinsdale,  who  was  a  student  at  Princeton  from  1852 
to  1856,  gives  the  following  reminiscences  of  the  mathematical  teaching 
there :  "  The  requirements  for  admission  were  geometry — four  books 
of  Euclid,  algebra  through  quadratics.  The  text-book  in  algebra  dur- 
ing the  Freshman  year  was  Hackley's.  The  fact  was,  that  but  few  who 
entered  were  fully  prepared,  and  therefore  we  had  a  rapid  review  of 
the  subjects  ah  initio,  finishing  Hackiey  the  first  year.  In  the  Sopho- 
more year  we  finished  Euclid's  geometry,  also  surveying  and  naviga- 
tion (elementary).  Both  subjects  were  taught  in  a  special  way  by  Prof. 
John  T.  Duf&eld,  whose  syllabus  taken  down  from  dictation  was  a 
marvel  of  clearness.  The  notes  of  that  syllabus  I  have  with  me.  It 
has  never  been  printed.  The  Junior  class  studied  Young's  Analytical 
Geometry  and  Conic  Sections.  The  first  half  of  the  year  we  were  taught 
by  Professor  Duffield,  the  last  half  by  Professor  Alexander,  who  had 
the  chair  of  physics  and  astronomy. 

"  In  the  Senior  class  mathematics  was  taught  by  Professor  Alexander, 
a  gentleman  of  marked  ability  in  the  higher  branches  of  his  depart- 
ments. He  used  no  text-book  in  either  department.  Both  subjects 
were  taught  orally.  An  elaborate  compendium  of  mathematical  physics 
'was  dictated  to  the  class  by  the  professor,  accompanied  by  explanations 
of  formulae  and  experimental  illustrations.  The  same  way  was  adopted 
by  the  professor  in  teaching  the  mathematics  of  astronomy.  His  sylla- 
bus in  that  department,  however,  was  '  printed,  not  published,'  for  tho 
use  of  the  class. 

*  Catalogue  of  the  College  of  New  Jersey,  1888-89,  p.  42. 
881— No.  3 11 


162  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

"Professor  Alexander  bad  a  distinguislied  repatation  among  his 
confreres.  Professor  Peirce,  of  Cambridge,  spoke  of  him  repeatedly 
in  public  lectures  as  the  'Kepler  of  the  nineteenth  century,'  always  in 
connection  with  his  theory  as  to  the  asteroids,  accompanied  by  mathe- 
matical demonstrations  that  they  once  formed  one  wafer-shaped  planet 
which,  '  somewhere,  somehow,'  was  shattered  into  fragments." 

Eev.  Horace  G.  Hinsdale  says : *  "He  [Alexander]  pushed  his  re- 
searches into  the  depths  of  mathematical  and  astronomical  science, 
availing  himself  of  his  acquaintance  with  the  principal  languages  of 
Europe.  He  printed  for  the  use  of  his  students  treatises  on  ratio  and 
proportion,  differential  calculus,  and  astronomy.  He  was  unselfish  in 
his  devotion  to  the  interests  of  the  college  and  the  advancement  of 
learning.  He  aroused  the  admiration  of  his  pupils  by  the  evident  ex- 
tent of  his  knowledge  and  his  ardor  in  imparting  it,  although  it  must 
be  said  that  he  often  became  so  profoundly  interested  in  setting  forth 
the  philosophy  of  mathematics  as  to  forget  that  their  acquaintance  with 
the  subject  was,  of  necessity,  far  less  than  his  own,  and  so  to  outrun 
their  ability  to  follow  and  comprehend  him.  The  closing  lectures  in  his 
course  in  astronomy,  in  which  he  discussed  the  nebular  hypothesis  of 
Laplace,  were  characterized  by  a  lofty  and  poetic  eloquence,  and  drew 
to  his  class-room  many  others  than  the  students  to  whom  they  were 
addressed.  Even  ladies  from  the  village  and  elsewhere — so  far  did  the 
traditional  conservatism  of  Princeton  give  way  before  a  wholesome  pres- 
sure---invaded  Philosophical  Hall." 

Professor  Young  says :  "  He  was  familiar  not  only  with  the  ordinary 
range  of  mathematical  reading,  but  with  many  works  of  higher  order. 
He  had  large  portions  of  the  Mecanique  Celeste  almost  at  his  finger's 
ends,  and  was  well  acquainted  with  the  works  of  Newton,  Euler,  and 
Lagrange." 

As  was  the  case  with  all  college  professors  in  former  years,  and  is 
still  true  with  most  of  them,  Professor  Alexander's  time  and  strength 
were  so  consumed  by  the  routine  duties  of  the  office,  that  little  remained 
for  anything  else.  Still  he  accomplished  a  great  deal.  He  published 
articles  in  various  scientific  journals,  and  presented  a  large  number  of 
papers,  orally,  before  scientific  societies ;  and  the  only  record  of  these 
communications  which  we  now  have  is  a  mere  notice  or  a  brief  abstract 
of  a  paper  read  on  such  and  such  a  date. 

In  1848  he  read  before  the  American  Academy  for  the  Advancement 
of  Science  a  paper  on  the  Fundamental  Principles  of  Mathematics. 
Prof.  C.  A.  Young  says  of  it :  "  It  is  an  interesting,  suggestive,  and  elo- 
quent essay.  The  subject  gives  the  author  an  opportunity  to  indulge 
his  inherited  Scotch  love  for  metaphysics  and  hair-splitting  distinctions, 
and  he  finds  in  it  also  opportunity  for  imagination  and  poetry  to  an 
extent  which  makes  the  paper  almost  unique  among  mathematical  dis- 
quisitions." 

*Quoted  by  Prof.  C.  A.  Young  in  hia  memoir. 


INFLUX   OF   FRENCH   MATHEMATICS.  163 

Professor  Alexander  was  an  astronomer,  but  his  special  forte  was  not 
that  of  the  observer.  In  fact,  he  had  no  adequate  instruments  or  ob- 
servatory. Long  did  he  labor  to  secure  a  good  observatory  for  the 
college,  and,  at  last,  in  1882,  a  great  telescope  was  pointed  toward  the 
stars.  "  There  was  something  pathetic  in  his  exclamations  of  satisfac- 
tion and  delight,  for  the  great  instrument,  so  long  dreamed  of,  had  only 
come  too  late  for  him  to  use  it." 

In  1876  Alexander  was  made  professor  emeritus,  and  Charles  Greene 
Eockwood  became  connected  with  the  mathematical  department.  Pro- 
fessor Eockwood  was  graduated  at  Yale  in  1864,  and,  before  going  to 
Princeton,  was  professor  of  mathematics  in  Bowdoin  and  Eutgers  suc- 
cessively. He  has  acquired  reputation  by  his  studies  of  earthquakes, 
and  has  contributed  articles  on  vulcanology  and  seismology  to  the  Ee- 
ports  of  the  Smithsonian  Institution,  1884-86. 

From  1878  to  1883  Dr.  G.  B.  Halsted  was  a  teacher  in  mathematics; 
until  1881  as  tutor,  and  from  that  time  on  as  instructor  in  postgraduate 
mathematics. 

The  present  mathematical  corps  consists  of  Professors  John  Thomas 
Duffield,  C.  G.  Eockwood,  H.  B.  Fine,  and  Tutor  H.  D.  Thompson.  C. 
A.  Young  is  the  successor  to  Alexander  as  professor  of  astronomy. 

The  conservatism  of  Princeton  College  is  noticeable  in  some  features 
of  the  mathematical  instruction.  Euclid  has  been  retained  as  a  text-book 
to  the  present  day.  Todhunter's  edition  has  been  used  now  for  many 
years.  Until  recently  Loomis's  text-books  were  used  largely,  though 
not  exclusively.  In  the  academical  under-graduate  department  the  fol- 
lowing mathematics  were  taught  in  1881 :  Freshman  year,  Eay's  Uni- 
versity Algebra,  Todhunter's  Euclid,  and  Mensuration ;  Sophomore  year, 
Loomis's  Plane  Trigonometry,  Navigation,  Surveying,  Spherical  Trigo- 
nometry, and  Analytical  Geometry;  Junior  and  ;S^emor  years,  Analytical 
Geometry  of  two  and  three  dimensions,  and  Calculus.  Under  Professor 
Duffield  oral  instruction  is  made  prominent.  It  might  be  more  correct 
to  say  that  mathematics  is  taught  by  him  "  mainly  by  lectures — the 
text-books  being  used  by  way  of  reference,  and  as  furnishing  examples 
for  practice."  "  The  students  are  required  to  take  notes  of  the  lectures 
and  submit  their  note-books  for  examination  at  the  end  of  each  term." 
Until  quite  recently  electives  were  introduced  very  sparingly.  At  pres- 
ent all  studies  are  prescribed  during  the  first  two  years ;  mathematics 
is  elective  during  the  last  two  years. 

Modern  higher  mathematics  was  first  introduced  in  Princeton  Col- 
lege by  Dr.  G.  B.  Halsted.  His  examination  papers  on  quaternions, 
determinants,  and  modern  higher  algebra,  are  the  first  ones  that  have 
ever  been  set  at  Princeton.  One  feature  of  the  mathematical  instruc- 
tion at  this  institution  that  has  been  in  vogue  during  the  last  ten  years 
(perhaps  longer)  is,  we  think,  to  be  recommended  for  more  general 
adoption.  Considerable  attention  is  given  to  the  study  of  the  history 
of  mathematics.    The  writer  has  before  him  examination  papers,  writ- 


164  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

ten  in  answer  to  questions  set  by  Halsted  in  1881.*  From  the  answers 
we  infer  that  questions  like  these  have  been  asked :  Who  wrote  the 
iirst  algebra  that  has  come  down  to  us  ?  What  was  its  nature  ?  What 
part  did  the  Hindoos  play  in  the  development  of  algebra  ?  Its  growth 
during  the  Eenaissauce *?    The  laws  underlying  ordinary  algebra?  etc. 

The  present  mathematical  coarse,  according  to  the  catalogue  of 
1888-89,  is  as  follows : 

For  admission  to  the  academical  department  of  the  college,  the 
mathematical  requirements  are :  "  Arithmetic,  including  the  metric 
system ;  algebra,  through  quadratic  equations  involving  two  unknown 
quantities — ^including  radicals,  and  fractional  and  negative  exponents  ; 
geometry,  the  first  and  second  books  of  Euclid,  or  an  equivalent — that 
is,  the  propositions  in  other  text-books  relating  to  the  straight  line  and 
rectilinear  figures,  not  involving  ratio  and  proportion." 

Studies  in  the  academical  department :  "  In  the  Freshman  year  there 
are  two  exercises  a  week  during  the  first  and  second  terms,  in  algebra, 
and  two  exercises  a  week  during  the  third  term,  in  plane  trigonometry, 
under  Professor  Fine;  in  geometry  there  are  two  exercises  a  week 
throughout  the  year,  under  Mr.  Thompson.  The  text-book  in  algebra 
is  Wells's  University  Algebra,  to  be  supplemented  by  a  course  on  the 
theory  of  equations,  by  the  professor.  Loomis's  Trigonometry  is  the 
text-book  in  trigonometry.  Euclid  is  used  as  the  text  book  in  geometry 
because  of  its  historical  associations  and  its  decided  superiority  for  the 
purpose  of  mental  discipline  to  any  modern  text-book.  The  first  six 
and  the  eleventh  books  of  Euclid  are  supplemented  by  a  course  in  solid 
and  spherical  geometry.  Since  a  thorough  knowledge  of  geometry  and 
familiarity  with  its  more  important  propositions  can  be  obtained  only 
by  extended  practice  in  the  demonstration  of  theorems  and  problems 
not  contained  in  the  text-book,  this  exercise  occupies  a  prominent 
place  in  the  course  of  instruction. 

"  The  Sophomore  class  has  three  exercises  a  week  throughout  the 
year  in  mathematics,  under  Professor  Duffield.  For  the  first  term  the 
studies  are  analytical  trigonometry,  mensuration,  and  navigation ;  for 
the  second  and  third  terms,  surveying,  spherical  trigonometry,  analyt- 
ical geometry,  and  the  elements  of  the  differential  calculus. 

"  In  the  Junior  year  mathematics  is  an  elective  study.  The  class 
has  two  exercises  a  week  throughout  the  year,  under  Professor  Duf- 
field. For  the  first  and  second  terms  the  studies  are  analytical  geom- 
etry and  the  differential  calculus ;  for  the  third  term,  the  integral  cal- 
culus. Loomis's  Trigonometry  is  the  text-book  during  the  first  and  Sec- 
ond terms  of  the  Sophomore  year.  Bowser's  Analytical  Geometry  and 
Calculus  during  the  third  term  Sophomore  and  Junior  year — supple- 
mented largely  by  oral  instruction,  and  numerous  exercises  in  addition 
to  the  examples  for  practice  of  the  text-books. 

*  One  of  these  vras  written  by  H.  B.  Fine,  now  assistant  professor  of  mathematics 
at  Princeton;  another  by  A.  L.  Kimball,  now  associate  professor  of  physics  at  the 
Johns  Hopkins  University. 


/ 

INFLUX  OP  FKENCH  MATHEMATICS.  165 

"The  Senior  class  in  mathematics  (elective)  lias  two  exercises  a  week 
throughout  the  year,  under  Professor  Pine.  The  course  for  the  current 
year  is  analytical  geometry  of  three  dimensions,  differential  and  inte- 
gral calculus.  Williamson's  text-books  on  the  calculus  are  used,  sup- 
plemented by  lectures  on  determinants,  differentiation  and  integration 
of  functions  of  the  complex  variable,  definite  integrals." 

In  1873  was  founded,  as  a  branch  of  Princeton  College,  a  scientific 
school  called  the  '^  John  0.  Green  School  of  Science."  Its  courses  lead 
to  the  degree  of  bachelor  of  science.  Two  years  later  a  course  in  civil 
engineering  was  organized  in  this  school.  The  mathematics  in  the 
scientific  school  is  taught  by  Professor  Eockwood.  The  course  is 
framed  so  as  to  supply  the  necessary  foundation  in  knowledge  and 
training  for  the  later  studies  of  physics  and  mechanics,  and  especially 
finds  its  natural  continuation  in  the  applied  mathematics  of  the  course 
in  civil  engineering.  Constant  blackboard  practice  is  a  prominent 
feature  of  the  instruction.  Euclid  is  supplanted  by  Chauvenet's  Geom- 
etry. Other  text-books  used  are  Wells's  Algebra,  Bowser's  Analytical 
Geometry  and  Calculus.  The  calculus  is  begun  at  the  end  of  the 
Sophomore  year  and  then  finished  in  the  Junior.  With  the  geometry, 
which  is  illustrated  by  models,  is  combined  a  thorough  course  in  men- 
suration and  an  introduction  to  the  elements  of  modern  geometry. 
Thus,  a  synthetic  course  in  conic  sections  is  made  to  precede  analytical 
geometry — an  idea  highly  to  be  recommended.  Calculus  is  required 
of  all  students  in  the  scientific  department.  More  advanced  studies 
in  pure  mathematics  are  elective. 

Descriptive  Geometry  is  taught  by  Professor  Willson  from  Warren's 
treatise. 

In  addition  to  the  college  courses,  there  are  at  Princeton  Univercity 
courses  leading  to  the  degrees  of  master  and  doctor. 

Post-graduate  mathematics  have  been  taught  since  1881. 

"The  University  courses  this  year  (1888-89)  are  in  differential  equa- 
tions, in  the  theory  of  functions,  and  in  higher  algebraic  curves  and  sur- 
faces. They  are  based  on  the  treatises  of  Forsytli  and  Boole,  Hermite 
and  Clebsch  and  Gordan,  and  Salmon  and  Clebsch,  respectively.  Pro- 
fessor Fine  conducts  these  courses. 

DARTMOUTH  COLLEGE. 

In  1833  Ira  Young  succeeded  Ebenezer  Adams  as  professor  of  math- 
ematics and  natural  philosophy.  His  father  was  a  carpenter,  which 
trade  he  followed  till  he  attained  his  majority.  He  early  manifested 
much  mechanical  ingenuity.  At  twenty-one  he  began  a  course  pre- 
paratory to  entering  college,  and  graduated  at  Dartmouth  in  1828.  He 
served  in  the  college,  first  as  tutor,  then  as  professor,  until  his  death  in 
1858.    He  is  said  to  have  been  an  admirable  teacher. 

From  1838  to  1851  Stephen  Chase  was  a  professor  of  mathematics  at 
Dartmouth.    He  was  a  graduate  of  this  college.    While  he  was  pro- 


166  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

fessor  he  published  an  algebra.  An  old  alumnus  speaks  of  him  as  a 
teacher,  ^'  the  light  of  whose  genius,  as  it  gleams  through  one  of  our 
text- books,  yet  lingers  in  our  halls." 

The  catalogue  for  1834  shows  that,  since  1828,  a  remodeling  of  the 
college  course  had  taken  place.  There  were  now  four  departments, 
viz,  the  classical,  mathematical  and  physical,  rhetorical,  and  the  depart- 
ment of  intellectual  and  moral  philosophy.  The  Freshmen  in  the  math, 
ematical  and  physical  department  studied  Playfair's  Euclid,  reviewed 
Adams's  Arithmetic,  and  commenced  Day's  Algebra  during  the  first 
term ;  continued  Day's  Algebra  in  the  second  term ;  and  completed 
Euclid  in  the  third. 

The  Sophomores  continued  Day's  Algebra,  devoting  their  attention  to 
applications  to  geometry  and  logarithms.  They  then  took  up  plane 
trigonometry  and  its  applications.  During  the  second  term  Bridge's 
Conic  Sections  and  Curvature,  and  Playfair's  Spherical  Geometry  and 
Trigonometry  occupied  their  attention.  They  began  also  Bezout's 
Calculus,  which  was  finished  during  the  third  term.  The  Juniors  pur- 
sued Olmsted's  Natural  Philo  sophy,  Day's  Mathematics  (heights  and 
distances,  and  navigation),  Olmsted's  Hydrostatics,  and  Astronomy. 
The  Seniors  had  no  mathematics,  according  to  catalogue. 

The  next  year  (1839)  indicates  several  changes.  Legendre's  Geometry 
and  Bourdon's  Algebra  displaced  old  Euclid  and  Day's  Algebra.  Da- 
vies'  Analytical  Geometry  and  Calculus  were  also  used.  The  influence 
of  the  Military  Academy  at  West  Point  was  now  beginning  to  be  felt  at 
Dartmouth. 

Eegarding  the  mathematical  teaching  at  this  time,  John  M.  Ordway, 
professor  of  applied  chemistry  and  biology  at  the  Tulane  University  of 
Louisiana,  writes  us  as  follows:  "  When  I  entered  Dartmouth  College 
in  1840,  the  Freshmen  were  instructed  in  algebra  and  geometry  by  a 
tutor.  We  used  Davies'  Bourdon's  Algebra  and  Davies'  Legendre's 
Geometry.  In  the  Sophomore  year  we  studied  Davies'  Surveying,  and 
Plane  and  Spherical  Trigonometry,  Davies'  Analytical  Geometry  and 
Davies'  Calculus.  The  instruction  was  given  by  Professor  Stephen 
Chase,  an  excellent  scholar,  but  a  somewhat  peculiar  man.  He  showed 
very  little  mercy  to  the  duller  students,  and  hence  was  not  very  popular. 
The  analytical  geometry  and  calculus  had  not  been  introduced  many 
years,  and  it  was  a  sort  of  traditional  idea  of  the  classes  that  preceded 
ours,  that  these  subjects  were  very  hard.  We,  however,  did  not  per- 
petuate this  tradition,  for  our  class  as  a  whole  did  not  find  these  higher 
mathematics  so  very  difficult.  We  had  some  field  exercises  in  surveying 
and  leveling.  The  professor  went  out  first  with  half  a  dozen  chosen  stu- 
dents of  the  class,  and  they  afterward  went  out  with  their  respective  sec- 
tions of  the  class.  Before  our  time  there  had  been  some  solemn  Durn- 
ings  of  the  mathematical  text-books  at  the  end  of  the  year,  but  we  had 
no  such  nonsense  while  I  was  in  college. 

"  Professor  Chase  also  gave  the  instruction  in  physics,  which  was 


INFLUX  OF  FRENCH  MATHEMATICS.  167 

quite  mathematical.  He  and  Professor  Young,  the  father  of  the  present- 
Professor  Charles  Young,  of  Princeton,  had  planned  and  partly  written 
a  work  on  physics,  in  which  the  demonstrations  were  to  be  made  by  the 
calculus  and  analytical  geometry ;  but  meanwhile  Professor  Olmsted 
published  his  iSTatural  Philosophy,  and  as  a  matter  of  courtesy  they 
dropped  their  own  work  and  introduced  the  poorer  one  of  Olmsted.* 
Olmsted  used  the  common  geometry  and  algebra,  and  his  book  was 
rather  old-fashioned  and  contained  some  absurd  errors.  There  was  one 
question  in  the  book,  'If  the  pebble  that  David  threw  weighed  2  ounces 
and  Goliath  weighed  800  pounds,  with  what  velocity  must  the  stone 
have  moved  to  prostrate  the  giant?'  The  answer  given  was  (about) 
2,800  feet  per  second,  or  greater  than  that  of  a  cannon  ball.  The  pro- 
fessor called  me  up  on  this  question,  in  the  recitation,  and  asked  me  if 
I  saw  any  absurdity  in  the  matter.  I  told  him  yes,  the  answer  should 
have  been  2,800,  and  not  2,800  feet  per  secoud.  Then  the  professor 
went  on  to  explain  that  Goliath  must  have  had  a  skull  that  would  be 
penetrated  by  a  stone  moving  with  much  less  velocity.  He  had  entirely 
overlooked  the  mathematical  absurdity  of  getting  a  concrete  answer  out 
of  mere  abstract  numbers.  I  went  to  him  after  recitation  to  explain  my 
idea  more  fully,  and  told  him  that  had  Mr.  Olmsted  been  a  Frenchman 
he  would  have  made  the  answer  2,800  meters  per  second,  and  that 
would  have  been  just  as  correct,  or  2,800  miles  would  have  done  just  as 
well.  This  he  acknowledged,  but  seemed  never  to  have  thought  of  it 
before,  the  physiological  absurdity  having  shut  out  from  his  perception 
the  mathematical  error. 

"  While  Professor  Chase  gave  the  mathematical  teaching  of  physics, 
Professor  Young  lectured  on  the  subject  with  the  help  of  a  very  good 
set  of  apparatus. 

"In  the  Junior  year  Professor  Young  taught  astronomy,  using  Olm- 
sted's Astronomy  for  a  text-book.  This  work  was  better  than  the 
physics,  but  it  rejected  the  calculus,  which  would  have  made  many  of 
the  demonstrations  much  plainer.  Professor  Young  was  an  excellent 
teacher  and  was  very  popular.  He  could  be  severe  enough,  but  it  was 
in  a  quiet,  dry  way  that  was  not  offensive.  He  would  call  up  a  fellow 
who  had  not  studied  the  lesson  well  and  put  several  questions,  receiv- 
ing the  wrong  answers  without  any  sign  of  surprise  or  demur,  and  finally 
say,  '  The  reverse  is  true,'  and  call  up  another  man. 

"  We  had  some  astronomical  instruments,  but  with  the  exception  of 
the  telescope  very  few  of  the  students  ever  used  any  of  them.  Our 
examinations  in  those  days  were  all  oral.  They  were  held  in  the  pres- 
ence of  a  committee  of  old  graduates  summoned  to  Hanover  for  the 
purpose,  their  expenses  being  paid  by  the  college.    These  old  fellows 

*  Olmsted's  Introduction  to  Natural  Philosophy  was  published  in  1831 ;  Young  was 
elected  professor  in  1833  and  Chase  in  1838.  While  Young  and,  we  believe,  also 
Chase,  served  as  tutors  before  they  were  appointed  professors,  it  is,  nevertheless,  not 
likely  that  reference  can  be  had,  in  the  above,  to  the  first  edition  of  Olmsted's  work 
on  natural  philosophy. 


168  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

were  rather  rusty  sometimes  and  gave  the  boys  some  amusement  by 
their  occasional  old-time  questions.  The  examinations  were  really  a 
farce,  though  the  results  were  counted  in  with  the  rest  of  the  marks. 
It  was  rather  funny  to  see  how  some  fellows  who  had  been  rated  very 
low  all  the  year  would  be  made  out  by  the  examiners  to  be  among  the 
very  best." 

In  1849  Chase's  Algebra  appeared,  and  began  to  be  used  at  Dart- 
mouth. Three  years  later  Loomis's  series  was  introduced,  excepting 
his  Algebra. 

In  1851  Chase  was  succeeded  by  John  Smith  Woodman,  of  the  class 
of  1842.  After  graduation  he  taught  school  in  Charleston,  S.  C,  after- 
ward made  a  tour  through  Europe  on  foot,  then  studied  and  practiced 
law,  and  finally  was  elected  professor  at  his  alma  mater. 

In  1854  James  Willis  Patterson,  of  the  class  of  1848,  became  professor 
of  mathematics.  He  had  previously  been  tutor  two  years.  From  1859 
till  1865  he  was  professor  of  astronomy  and  meteorology.  He  after- 
ward entered  politics,  was  elected  to  the  Legislature  and  finally  to  the 
United  States  Senate. 

About  the  middle  of  the  present  century  attempts  were  made  to  or- 
ganize a  system  of  education  based  chiefly  upon  the  pure  and  applied 
sciences,  modern  languages,  and  mathematics.  Of  this  class  were  the 
scientific  schools  connected  with  colleges,  such  as  the  Lawrence  Scientific 
School  at  Harvard,  the  Sheffield  Scientific  School  at  Yale,  the  School 
of  Mines  at  Columbia,  and  the  Chandler  Scientific  School  at  Dart- 
mouth. These  schools  have  done  efficient  work  and  supplied  a  long- 
felt  want. 

The  Chandler  Scientific  School  was  established  in  1851.  The  instruc- 
tion was  designed  to  be  "in  the  practical  and  useful  arts  of  life,  com- 
prised chiefly  in  the  branches  of  mechanics  and  civil  engineering."  At 
first  J.  W.  Patterson  is  given  in  the  catalogue  as  Chandler  professor  of 
mathematics,  but  Professor  Woodman  was  the  one  who  labored  longest 
in  this  school.  He  taught  in  it  from  its  establishment,  became  profes- 
sor of  civil  engineering  in  1856,  and  was  practically  at  the  head  of  it. 
He  retained  those  posts  until  his  death  in  1871. 

The  mathematical  course  in  this  school  was  low  at  the  beginning. 
Loomis's  books  were  used,  also  Puissant's  Math3matics.  Descriptive 
geometry,  shades,  and  shadows  were  also  introduced. 

In  the  catalogue  of  1865,  Robinson's  series,  from  the  Algebra  to  the 
Difierential  and  Integral  Calculus,  is  given.  In  1866  Church's  Analyt- 
ical Geometry  and  Calculus  were  studied  in  the  Chandler  Scientific 
School.  Two  years  later  the  college  dropped  Eobinson's  series  and 
returned  to  Loomis's. 

Since  1870  the  text-books  used  have  been,  in  algebra,  Olney,  Quimby ; 
in  geometry  and  trigonometry,  Olney  ',  in  analytical  geometry,  Loomis, 
Church,  Olney;  in  calculus.  Church,  Olney;  in  analytical  mechanics. 
Peck,  Wood  j  in  descriptive  geometry,  Church  j  in  quaternions,  Hardy. 


INFLUX  OF  FRENCH  MATHEMATICS.  169 

The  terms  of  admission  to  the  college  were,  in  1828,  arithmetic,  alge- 
bra through  simple  equations;  in  18il,  the  same;  in  1864,  the  same, 
with  the  addition  of  two  books  of  (Loomis's)  geometry ;  in  1886  and  for 
some  years  previous,  ail  of  plane  geometry  was  required ;  in  1888,  arith- 
metic, including  the  metric  system,  algebra  to  quadratics,  and  plane 
geometry. 

The  college  offers  now  two  courses,  one  leading  to  the  degree  of 
bachelor  of  arts,  the  other  (the  Latin-scientific  course)  to  the  degree  of 
bachelor  of  letters.  The  course  of  study  for  the  year  1888-89  is  as 
follows : 

In  the  Prescribed  Courses,  I  is  in  each  case  an  advanced  division  for  students  judged  to  be  qualified  to 

pursue  a  more  extended  course. 

PEESCEIBED  COURSES. 
Freshman  Year. 

1.  I.  Algebra,  including  Theory  of  Equations  (Quimby).    Sixty-five  lioura. 
II.  Algebra.    Sixty-five  hours. 

2.  I.  Solid,  with  advanced,  Geometry  (Olney).     Forty-five  hours. 
II.  Solid  Geometry.    Forty-five  hours. 

3.  I.  Plane  trigonometry  (Olney),  including  applications  to  Surveying ;  Spheri- 

cal Trigonometry,    Sixty-two  exercises  {including  ten  exercises  of  field  work 
of  three  hours  each). 
II.  Same  as  3, 1,  omitting  Spherical  Trigonometry. 
Sophomore  Year. 

4.  I.  Analytic  Geometry  (Olney).    Forty  hours. 

II.  Spherical  Trigonometry  and  Conic  Sections.     Forty  hours. 

5.  Surveying  with  field  work  and  plotting.     Eighty-seven  hours. 

Course  5  is  open  only  to  students  of  the  Latin- Scientific  Course, 

6.  Descriptive  Geometry ;  Drawing.    Sixty  hours. 

Course  6  is  open  only  to  students  of  the  Latin-ScieniiUo  Course. 

ELECTIVE  COURSES. 

7.  a.  Differential  Calculus.    ?  Applications    to    Analytic    Geometry.     Lectures, 
b.  Integral  Calculus.         5     Ninety-four  hours. 

Course  7,  a  and  b,  is  elective  tvith  French  2  followed  "by  Mathematics  8. 

8.  Elementary  Mechanics  (Wood).     Fifty  hours. 

Course  8  preceded  iy  French  2  is  elective  with  MatheDiatics  7,  a  and  b. 
Junior  Year. 

9.  Analytic  Mechanics ;  Lectures.     Sixty  hours. 

Course  9  is  open  only  to  students  ivho  have  completed  Course  7,  and  is  elective  with 
Latin  and  GreeTc. 
10.  Descriptive  Geometry;  Shades,  Shadows,  and  Perspective  (Church).    Forty- 
four  hours. 
Course  10  is  elective  iviih  Latin,  Greek,  German,  Physics. 

The  minimum  amount  of  mathematics  on  which  a  degree  can  be  ob« 
tained,  is  a  course  ending  with  spherical  trigonometry  and  conic  sec- 
tions.   Analytic  geometry  is  not  necessary. 

The  course  in  pure  mathematics  in  the  Chandler  Scientific  School  is 
much  the  same  as  the  above.  The  catalogue  mentions  in  that  depart- 
ment Olney  as  the  text-book  in  calculus  and  Peck  as  that  in  analytical 
mechanics. 


170  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

In  1860,  John  E.  Varney  was  appointed  professor  of  mathematics, 
and  served  for  three  years.  During  the  next  six  years  John  E.  Sinclair 
filled  this  position.  In  1872  F.A.  Sherman,  the  present  professor  of 
mathematics  in  the  Chandler  School  of  Science,  was  elected.  From 
1872  to  1878  C.  F.  Emerson  was  connected  with  the  mathematical  de- 
partment. Since  then  he  has  occupied  the  chair  of  natural  philosophy 
and  has  devoted  his  energies  chiefly  to  the  development  of  the  physical 
laboratory.  P.  H.  Pettee  has  been  professor  of  mathematics  since  1877, 
and  is  now  teaching  mathematics  and  engineering  in  the  New  Hamp- 
shire Agricultural  Experiment  Station,  which  is  a  branch  of  Dartmouth 
College.  At  present  T.  W.  D,  Worthen  is  associate  professor  of  math- 
ematics in  the  college. 

Since  1878  Arthur  S.  Hardy  has  been  the  head  of  the  mathematical 
department  at  Dartmouth.  He  is  professor  of  mathematics  and  of  civil 
engineering.  Previous  to  the  above  date  he  held  the  professorship  of 
civil  engineering  in  the  Chandler  Scientific  School.  Professor  Hardy 
was  graduated  at  West  Point  in  1869.  For  three  years  he  was  professor 
of  civil  engineering  at  Iowa  College.  He  then  spent  one  year  in  study 
at  the  Ecole  imp^riale  des  ponts  et  chaussees  in  Paris,  and  on  his  return 
went  to  Dartmouth.  In  1881  appeared  his  Elements  of  Quaternions,  the 
first  American  book  on  this  subject.  It  is  elementary  and  well  adapted 
for  use  of  those  students  in  our  colleges  who  may  desire  to  know  some- 
thing of  the  wonderful  researches  of  Sir  William  Eowan  Hamilton.  A 
neat  little  publication  of  much  interest  is  Professor  Hardy's  transla- 
tion from  the  French  of  Argand's  Imaginary  Quantities.  He  published 
also  Kew  Methods  of  Topographical  Surveying,  1884.  Professor  Hardy 
possesses  two  qualifications  that  are  rarely  combined ;  he  is  a  successful 
mathematician  and  also  a  successful  novelist. 

BOWDOIN  COLLEGE.* 

In  1825  William  Smyth  became  adjunct  professor  of  mathematics, 
and  in  1828  was  given  the  full  chair,  which  he  held  until  his  death  in 
1868.  He  was  an  alumnus  of  thecollege.  After  his  graduation,  in  1822, 
he  studied  theology  at  Andover,  and  then  became  tutor  at  his  ahna 
mater.  He  was  led  to  abandon  Greek  and  take  the  department  of 
mathematics  as  an  instructor,  from  his  success  in  popularizing  algebra 
by  means  of  the  blackboard. 

The  introduction  of  the  blackboard  in  our  colleges  must  have  caused 
important  changes  in  the  methods  of  teaching  mathematics,  especially 
geometry.  Unfortunately  no  record  of  these  changes  has  been  pre- 
served except  at  one  or  two  institutions.  We  are  happily  able  to  quote 
the  following  account  of  its  introduction  at  Bowdoiu,  taken  from  the 
history  of  the  college,  written  by  A.  S.  Packard.  He  says  (p.  91)  that 
the  blackboard  was  introduced  by  "Proctor  (afterward  Professor) 

*We  are  indebted  for  the  information  herein  contained  chiefly  to  a  commanication 
from  Prof.  George  T.  Little,  of  Bowdoin  College. 


INFLUX   OF   FRENCH   MATHEMATICS.  171 

Smytli  in  1824.  That  novelty,  let  me  here  say,  made  a  sensation.  When 
he  had  tested  the  experiment  in  the  Sophomore  algebra,  and  with  great 
success,  a  considerable  portion  of  the  Juniors  requested  the  privilege  of 
reviewing  the  algebra  under  the  new  method  at  an  extra  hour — a  won- 
der in  college  experience  j  and  that  blackboard  experiment,  I  am  sure, 
led  to  his  appointment  as  assistant  professor  of  mathematics  a  year 
after.  Of  this  also  I  am  sure,  that  he  had  then  first  detected  a  math- 
ematical element  in  his  mental  equipment.    His  forte  had  been  Greek." 

Professor  Packard  gives  also  an  interesting  account  of  the  modes  of 
teaching  immediately  before  the  blackboard  came  to  be  used.  "The 
blackboard  caused  important  changes  in  the  manner  of  teaching  gen- 
erally, but  especially  in  the  mathematical  branches.  In  arithmetic,  a 
Freshman  study,  and  algebra,  to  which  we  were  introduced  at  the  open- 
ing of  the  Sophomore  year,  each  student  had  his  slate,  and  when  he 
finished  his  work  he  took  the  vacant  chair  next  the  teacher's  and  under- 
went examination  of  the  process  or  principle  involved.  In  geometry  we 
kept  a  MS.,  in  which  we  drew  the  figures  and  demonstrated  from  that. 
I  have  been  shown  a  very  neat  MS.  kept  at  Harvard  by  the  late  Dr. 
Lincoln,  the  father,  and  bearing  the  date  1800.  *  *  *  It  may  sur- 
prise my  hearers  that  I  professed  to  teach  the  algebra  of  the  Sopho- 
more class  in  Webber's  Mathematics — the  firsttutor,  I  believe,  to  whom 
the  duty  was  entrusted.  That  was  the  class  of  1824.  Franklin  Pierce, 
of  the  class,  in  his  earlier  years  of  college  life,  more  fond  of  fun  than 
of  surds  and  equations,  took  his  seat  by  my  side  for  a  quiz  with  his 
slate  and  solution  of  a  problem.  When  asked  how  he  obtained  a  cer- 
tain process ;  he  replied  very  frankly,  '  I  got  it  from  Stowe's  slate.' 
*  *  *  With  blackboard  such  transfers  are  less  easy.  *  *  *  It  will 
cause  more  surprise  that  conic  sections  in  Webber,  a  Junior  branch,  fell 
under  my  charge.  The  manner  of  reciting  was  simply  to  explain  the 
demonstration  in  the  text-book." 

In  1834  the  requirements  for  admission  were  increased  so  as  to  in- 
clude "  six  sections  of  Smyth's  Algebra."  These  six  sections  include 
nearly  the  entire  Algebra,  logarithms  and  ihe  binomial  theorem  being 
excluded.  In  1867  the  requirements  were  raised  so  as  to  read  arithme- 
tic, the  first  eight  sections  of  Smyth's  New  Elementary  Algebra  (to 
equations  of  the  second  degree),  and  the  first  and  third  books  of  Da- 
vies'  Legendre.  The  requisites  remained  practically  the  same  from 
1867  to  1887,  though  the  text-books  recommended  were  several  times 
changed.    Since  that  time  all  of  plane  geometry  has  been  required. 

The  calculus  first  appears  as  a  study  in  the  annual  catalogue  of  1830, 
the  notation  of  Leibnitz  being  then  used.  Fluxions  were  probably 
never  taught  at  Bowdoin. 

Professor  Smyth  became  an  exceedingly  able  teacher  and  gained 
celebrity  as  a  successful  writer  of  mathematical  text-books.  His  pub- 
lications were,  a  work  on  Plane  Trigonometry,  followed  by  his  Algebra, 
Analytical  Geometry  (1855),  and  Calculus  (1856).    All  of  these  passed 


172  TEACHING   AND    HISTOKY    OF   MATHEMATICS.  » 

tbrougb  repeated  editions  and  enjoyed  an  extensive  sale.  As  they  came 
from  the  press  they  took  the  place  of  the  Cambridge  Mathematics  at 
Bowdoin.  In  the  preparation  of  his  Algebra  he  followed  Bourdon  and 
Lacroix  as  models,  and  it  contains  many  of  the  excellences  and  some  of 
the  defects  of  these  works.  A  remarkable  feature  is  the  very  late  in- 
troduction and  explanation  of  negative  quantities.  They  appear  on 
page  89,  after  the  solution  of  simultaneous  linear  equation.  In  his  cal- 
culus he  uses  infinitesimals.  "As  a  logical  basis  of  the  Calculus,"  says 
he  (p.  229),  "  the  method  of  Kewton,  and  especially  that  of  Lagrange, 
has  some  advantage.  In  other  respects  the  superiority  is  immeasura- 
bly on  the  side  of  the  method  of  Leibnitz.^'  At  the  end  of  the  book  he 
very  briefly  explains  the  methods  of  li^^ewton  and  Lagrange.  A  few 
pages  are  also  given  to  the  "  Method  of  Yariations"  and  "Applications 
to  Astronomy." 

The  following  account  of  Smyth  and  his  works  is  taken  from  an  obitu- 
ary address  by  his  colleague,  Professor  Packard :  "  As  the  first  fruits, 
he  issued  a  small  work  on  Plane  Trigonometry,  availing  himself  of  the 
ingenuity  of  the  late  Mr.  L.  T.  Jackson,  of  this  town,  in  preparing  blocks 
on  a  novel  plan  for  striking  off  the  diagrams.  The  first  edition  of  his 
Algebra  from  the  press  of  Mr.  Griffin,  of  this  town,  appeared  in  1830, 
which  first  adapted  the  best  French  methods  to  the  American  mind,  re- 
ceived warm  commendation  from  Dr.  Bowditch,  and  was  adopted  as  a 
text-book  at  Harvard  and  other  institutions.  It  passed  through  several 
editions  and  then  gave  place  to  two  separate  works,  the  Elementary  Al- 
gebra and  the  Treatise  on  Algebra.  Then  followed  an  enlarged  edition 
of  the  Trigonometry  and  its  application  to  Surveying  and  Navigation, 
and  treatises  on  Analytic  Geometry  and  on  the  Calculus,  the  last  being 
so  clearly  and  satisfactorily  developed  and  with  so  much  originality  as  to 
receive  emphatic  approval  in  high  quarters,  particularly  from  the  late 
Professor  Bache." 

"  In  explanation  he  was  precise,  simple,  and  clear.  He  had  great  power 
of  inspiring  interest;  his  own  enthusiasm,  which  often  kindled,  espe- 
cially in  certain  branches  of  his  department,  at  the  blackboard,  being 
communicated  to  his  class.  Later  classes  will  carry  through  life  his 
setting  forth  of  what  he  termed  the  '  poetry  of  mathematics,'  as  exem- 
plified in  the  Calculus." 

Of  the  graduates  of  Bowdoin  during  Smyth's  time  who  distinguished 
themselves  in  the  mathematical  line,  we  mention  John  H.  C.  Coffin 
(class  of  1834),  who,  soon  after  graduation,  was  appointed  professor  of 
mathematics  in  the  U.  S.  Navy.  He  was  for  many  years  in  the  Naval 
Observatory  and,  in  1866,  took  charge  of  the  American  Ephemeris  and 
Nautical  Almanac. 

Professor  Smyth's  successor,  in  1865,  was  Charles  Greene  Eockwood, 
who  had  graduated  at  Yale  in  1864,  and  in  1866  received  the  degree  of 
Ph.  D.  When  he  left  Bowdoin  to  accept  a  position  at  Eutgers  College, 
in  1873,  Charles  Henry  Smith  took  his  place.    In  1SS7,  Professor  Smith 


INFLUX  OF  FRENCH  MATHEMATICS.  173 

was  succeeded  by  Prof.  William  Alboin  Moody,  the  present  incumbent 
of  the  chair  of  mathematics.  Professors  Eockwood  and  Smith  left  the 
college  with  the  reputation  of  able  and  skillful  teachers.  "  The  latter 
was,  in  my  judgment,  remarkably  successfal,"  says  Professor  Little,  "in 
securing  good  and  faithful  work  from  all."  The  writer  has  before  him 
a  report  on  geometry  by  Professor  Smith,  presented  to  the  Maine  Ped- 
agogical Society  in  1884,  and  containing  some  good  recommendations 
on  the  study  of  its  elements.  He  strongly  recommends  a  course  in  em- 
pirical geometry  of  the  sort  marked  out  by  G.  A.  Hill's  Geometry  for 
Beginners,  Mault's  Natural  Geometry,  and  Spencer's  Inventional  Geom- 
etry,  to  precede  the  course  in  demonstrative  geometry. 

Mathematics  have  never  been  taught  at  Bowdoin  by  lectures,  though 
the  instruction  has  been  frequently  supplemented  by  lectures.  Since 
1880  all  mathematics  have  been  elective  after  the  Sophomore  year  5 
since  1886,  all  after  the  Freshman  year.  An  elective  in  calculus,  not 
then  a  required  study,  was  offered  from  1870  to  1880.  In  the  year  1882- 
83  the  Freshmen  studied  Loomis's  Algebra,  and  Loomis's  Geometry  and 
Conic  Sections,  in  two  parallel  courses  during  the  first  two  terms ;  the 
third  term  of  the  year  being  given  to  Plane  Trigonometry  (Oluey).  The 
Sophomores  had  Olney's  Spherical  Trigonometry  during  the  first  term  5 
during  the  second  and  third  term  they  had  the  choice  between  analyti- 
cal geometry,  and  Latin  and  Greek.  Calculus  was  elective  for  Juniors, 
In  the  Senior  year  no  mathematics  were  offered.  The  text-book  in 
astronomy  was  JSTewcomb  and  Holden,  A  feature  in  this  mathematical 
course  to  be  recommended  is  that  analytical  geometry  is  preceded  by  a 
short  course  in  conic  sections  (treated  synthetically).  The  course  for  the 
year  1888-89  differs  from  the  preceding  in  this,  that  plane  geometry  is 
required  for  admission ;  that  Weutworth's  Algebra  has  taken  the  place 
of  Loomis's;  that  differential  and  integral  calculus  are  studies  in  the 
second  and  third  terms  of  the  Sophomore  year  5  that  an  advanced  course 
in  calculus  (Williamson's)  is  offered  during  the  first  two  terms  of  the 
Junior  year,  and  quaternions  during  the  third  term. 

GEOEaETOWN   COLLEaE.* 

From  1831  to  1879  Father  James  Carley  was  the  head  of  the  mathe- 
matical and  astronomical  department  at  Georgetown  College.  He  was 
born  in  Ireland,  October  25, 1796.  He  entered  the  Society  of  Jesus 
September  29, 1827,  and  came  here  in  1830.  In  1813  Father  Curley 
built  the  college  observatory.  Here  he  calculated,  from  his  observa- 
tions, the  longitude  of  Washington.  The  astronomers  at  the  U.  S. 
Naval  Observatory  had  found  a  longitude  differing  a  little  from  Father 
Cnrley's  result.    When,  however,  the  laying  of  the  Atlantic  cable 

*  For  what  material  we  possess  on  fche  teacMng  at  this  college  we  are  indebted  to 
the  kindness  of  Prof.  J.  F.  Dawsou,  S.  J.,  professor  of  physics  and  mechanics  at 
Georgetown  College. 


174  TEACHING   AND  HISTORY   OF  MATHEMATICS. 

brought  Washington  into  telegraphic  communication  with  Greenwich, 
it  was  found  that  Father  Curley's  calculation  was  the  correct  one. 
Since  1879  Father  Curley  has  not  been  able  to  teach ;  he  is  still  living 
at  Georgetown,  and  is  in  the  full  possession  of  all  his  faculties. 

Since  1839  Father  Curley  has  generally  had  two  assistants,  or  asso- 
ciates, in  mathematics.* 

Eev.  James  Clark  was  born  October  21,  1809.  He  entered  West 
Point  at  the  age  of  sixteen,  and  graduated  in  the  class  of  1829.  He 
served  in  the  Army  several  years.  In  1844  he  entered  the  Society  of 
Jesus,  and  came  to  Georgetown  in  1845.  In  1849  he  went  to  Worcester 
College,  Massachusetts,  then  recently  established,  but  remained  there 
only  one  year,  returning  to  Georgetown  in  1850.  From  1862  to  1867  he 
was  president  of  Worcester  College.  He  returned  to  Georgetown  in 
1867,  but  was  appointed  president  of  Gonzaga  College,  Washington,  in 
1869.  This  ofiBce  he  held  until  1875,  when  he  again  took  his  old  posi- 
tion in  Georgetown.  In  1879  Father  Clark  became  unable  to  teach, 
and  on  September  9, 1885,  he  died  at  Georgetown.  For  some  years  he 
taught  calculus  from  his  own  manuscript,  and  intended  to  publish  a 
text-book  but  for  some  reason  did  not  do  so. 

About  the  year  1848,  political  troubles  in  Europe  induced  a  consider- 
able emigration  to  America  of  some  of  the  most  able  members  of  the 
Society  of  Jesus,  and  the  faculty  of  Georgetown  College  was  increased 
by  a  considerable  accession  of  learning  and  talent.  We  mention  as  the 
most  conspicuous,  Fathers  Sestini  and  Secchi. 

Eev.  Benedict  Sestini  was  born  in  Italy,  March  20, 1816.  He  entered 
the  society  in  1836.  In  1847  he  was  astronomer  of  the  Eoman  Observa- 
tory. In  1848  he  was  compelled  to  leave  Italy  by  the  revolutionists, 
and  came  to  Georgetown.    He  taught  here  until  1857 ;  then  he  taught 

•  During  1839-'42,  1843-'45,  1847-'48,  and  1865-'67,  Rev,  James  Ward,  S.  J.,  gave  in- 
struction in  mathematics ;  1841-46,  Rev.  Thomas  Jenkins,  S.  J ;  1840-'41  and  1842-'43, 
Eev,  Augustine  Kennedy,  S.  J. ;  1844-'45,  Rev,  George  Fenwick,  S,  J. ;  1846-'47  and 
1869-'71,  Rev.  Joseph  O'Callaghan,  S.  J. ;  1848-'49,  Rev.  Angelo  Secchi,  S,  J.  ;  1849-'52, 
Rev.  Edward  McNerhany,  S.  J. ;  1852-54,  Rev.  Anthony  Vauden  Heuvel,  S.  J. ; 
l854-'60,  Rev.  John  Prendergast,  S.  J. ;  1860-'61,  1862-'63,  1870-'71,  1879-'P" ,  Rev.  C. 
Bahan ;  1861-'63  and  1871-74,  Rev,  G,  Strong,  S.  J. ;  1845-'49,  1850-'62,  an/  1875-'79, 
Rev,  James  Clark,  S,  J. ;  1848-'57,  and  1863-'69,  Rev.B.  Sestini ;  1863-'64,  Rev.  Aloy- 
sius  Varsi,  S.  J. ;  1864-'65,  Rev.  James  Major,  S,  J, ;  1867-'69,  Rev.  Antonio  Cichi,  S. 
J.  ;  18G9-'70,  Rev.  Patrick  Forhan,  S.  J. ;  1871-72,  Rev.  Patrick  Gallagher,  S.  J. ; 
1872-73,  Rev.  Jerome  Daugherty,  S  J. ;  1873-74,  Rev.  Edmund  Young,  S.  J.  :  1874- 
'78,  Rev.  J,  Ryan,  S.  J.  ;  1874-78,  Rev,  M,  O'Kane,  S.  J. ;  1878-'83,  Rev.  J.  R.  Rich- 
ards, S.  J.  ;  1879-'83,  Rev.  Henry  T.  Tarr,  S.  J. ;  188l-'86,  Mr.  Thomas  McLoughlin, 
S.  J. ;  1883-'84,  Rov.  Timothy  Brosnaham,  S.  J. ;  lS83-'84,  Rev.  John  O'Rourke,  S.  J. ; 
1884-'85,  Rev.  Edward  Devitt,  S  J.  ;  1884-'85,  Rev.  Thomas  Stack,  S.  J.  ;  1885-'88, 
Eev.  Samuel  H.  Frisby,  S.  J.  ;  1885-'87,  Mr.  Joseph  Gorman,  S.  J. ;  1887-'88,  Mr. 
David  Hearn,  S.  J. ;  1888-'—,  Rev.  John  Hagen,  S.  J.,  Rev.  John  Lehy,  S.  J.,  Mr. 
James  Dawson,  S.  J.,  Mr.  J.  Gorman,  S.  J. 

The  frequent  changes  in  the  corps  of  instructors  are  due  to  the  custom  of  the 
Society  of  Jesus.  "In  the  society  a  teacher  is  liable  any  year  to  be  sent  to  another 
college,  and  is  rarely  left  more  than  four  or  five  years  in  one  place." 


INFLUX  OF  FRENCH  MATHEMATICS.  175 

three  years  in  Gonzaga  College,  two  years  in  Worcester  College,  and 
one  year  in  Boston  College.  In  1863  he  returned  to  Georgetown,  where 
he  tanght  until  1869 ;  he  was  then  removed  to  Woodstock  College.  In 
1886,  advanced  in  years  and  broken  down  in  health,  he  was  sent  to 
Frederick,  to  the  novitiate  of  the  society,  where  he  still  remains  await- 
ing his  end.*  His  books  were  used  several  years  at  Woodstock  College 
(the  scholasticate  of  the  society).  At  one  time  they  were  in  rather  ex- 
tensive use,  but  at  present  they  have  gone  out  of  use  almost  com- 
pletely. 

Eev.  Benedict  Sestini  published  the  following  mathematical  works : 
A  Treatise  on  Analytical  Geometry,  Washington,  1852;  A  Treatise  on 
Algebra,  Baltimore,  1885;  Elementary  Algebra,  second  edition,  1855  (?) ; 
Elementary  Geometry  and  Trigonometry,  1856 ;  Manual  of  Geometrical 
and  Infinitesimal  Analysis,  Baltimore,  1871.  The  method  of  treatment 
of  the  various  subjects  in  these  works  is  not  entirely  conformable  to  that 
generally  in  vogue  in  this  country  at  the  time  of  their  publication.  The 
last  named  work  is  a  thin  volume  of  130  pages,  making  no  pretension 
of  being  a  complete  work  on  the  subject.  It  was  intended  primarily  for 
students  in  the  author's  own  classes  at  Woodstock  College,  in  Maryland, 
and  as  an  introduction  to  the  study  of  physical  science. 

With  Father  Sestini  came  Rev.  Angelo  Secchi,  the  astronomer.  He 
was  born  in  1818,  and  entered  the  society  in  1833.  He  was  compelled  to 
leave  Italy  in  1848,  on  account  of  the  revolution.  He  remained  at 
Georgetown  very  little  more  than  a  year.  In  1850  he  returned  to  Italy, 
and  was  placed  in  charge  of  the  Roman  Observatory,  where  he  labored 
until  his  death,  February  26,  1878. 

At  present  the  mathematical  course  consists  of  geometry,  plane  and 
spherical  trigonometry,  analytical  geometry,  differential  and  integral 
calculus,  mechanics,  and  astronomy.  Algebra  is  taught  in  the  prepar- 
atory department.  This  course  has  remained  practically  the  same  since 
1829,  except  that  the  time  given  to  mechanics  has  been  increased. 

Elective  studies  have  never  been  offered  at  the  college,  nor  has  the 
practice  of  lecturing  ever  been  in  vogue.  Since  1829  more  time  has 
been  given  to  mathematics  than  formerly.  About  the  year  1820  the 
Society  of  Jesus  adopted  a  new  ^^  ratio  studiorum,"  or  plan  of  studies, 
giving  to  mathematics  more  attention  than  had  hitherto  been  accorded 
to  them.  This  brought  about  the  change  at  Georgetown  in  1829.  The 
methods  of  the  Society  of  Jesus  have  been  strictly  adhered  to.  "  The 
professor  first  explains  the  lesson,  pointing  out  the  important  parts, 
the  proofs,  the  connection  with  other  parts  of  the  subject,  etc.,  and  giv- 
ing other  proofs  if  those  in  the  book  do  not  suit  him.  On  the  following 
day  he  calls  on  one  of  the  class  for  a  repetition ;  after  the  repetition 

*  Professor  Dawson  has  endeavored  to  find  out  something  about  the  early  life  and 
education  of  Father  Sestini,  but  with  no  success.  Father  Sestini  himself  can  not  give 
any  information  on  the  subject;  his  health  has  failed  very  much,  and  his  memory  can 
not  be  relied  upon. 


176  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

the  members  of  the  class  bring  forward  their  difficulties,  suggestions, 
etc.,  after  which  the  following  lesson  is  explained.  Problems  are  fre- 
quently given  to  test  the  knowledge  and  inventive  powers  of  the  stu- 
dents." 

Father  Sestini's  text-books  were  used  several  years.  They  were  re- 
placed by  those  of  Davies.  In  1860  Gummere's  surveying  was  intro- 
duced, and  the  Algebra,  Geometry,  and  Trigonometry  of  Davies ;  Sesti- 
ni's Analytical  Geometry  and  Calculus  were  retained.  In  1870  Sestini's 
Analytical  Geometry  and  Calculus  were  replaced  by  Davies'  Analytical 
Geometry  and  Church's  Calculus.  In  1872  and  1873  Loomis's  Analytical 
Geometry  was  used.  In  1874  Olney's  Algebra,  Trigonometry,  and  Calcu- 
lus were  introduced ;  Davies'  Geometry  and  Gummere's  Surveying  were 
retained.  In  1878  the  Algebra  and  Geometry  of  Loomis  were  used,  and 
in  1879  his  Trigonometry,  Analytical  Geometry,  and  Calculus.  Two 
years  ago  Wentworth's  series  was  introduced,  with  Taylor's  Calculus. 
Peck's  Mechanics  was  used  until  1881,  when  it  was  replaced  by  Dana's. 
In  calculus  the  notation  of  Leibnitz  has  been  employed  "  as  far  back  as 
we  have  any  records." 

At  the  college  observatory  no  work  has  been  done  for  some  years; 
but  in  January,  1889,  Eev.  John  Hagen,  S.  J.,  was  placed  in  charge  of 
the  observatory  and  will  make  regular  observations.  Father  Hagen 
was  formerly  at  Prairie  du  Chien,  Wis.  He  is  a  mathematician  of  con- 
siderable ability  and  has  contributed  articles  to  the  American  Journal 
of  Mathematics. 

COENELL  UNIVEHSITT.* 

When  Dr.  Andrew  D.  White  entered  upon  the  organization  of  Cornell 
University  and  the  selection  of  a  faculty,  the  first  professor  appointed 
was  Evan  William  Evans.  He  occupied  the  chair  of  mathematics  at 
Cornell  from  the  time  of  its  opening,  in  1868,  till  1872,  when  he  resigned 
on  account  of  failing  health.  Professor  Evans  was  a  native  of  Wales, 
came  to  this  country  with  his  parents  when  a  child,  was  graduated  at 
Yale  in  1851,  and  studied  theology  for  a  year.  He  then  became  princi- 
pal of  the  Delaware  Institute,  Franklin,  N.  Y.,  was  tutor  at  Yale  from 
1855  to  1857,  and,  later,  professor  of  natural  philosophy  and  astronomy 
in  Marietta  College,  Ohio,  where  he  remained  until  1864.  Before  en- 
tering upon  his  work  at  Cornell  University  he  was  occupied  for  three 
years  as  mining  engineer,  and  spent  one  year  in  European  travel.  He 
died  not  long  after  resigning  his  position  at  Cornell. 

In  the  same  year  that  Professor  Evans  was  selected  to  the  mathemat- 
ical chair,  Ziba  Hazard  Potter,  a  graduate  of  Hobart  College,  was  ap- 
pointed assistant  professor  of  mathematics.  This  position  he  held  for 
fourteen  years. 

"The  writer  is  indebted  to  the  kindness  of  Professor  Oliver  for  sending  annx3;i] 
reports  and  giving  information  on  the  mathematical  courses  3f  study  at  Cornell  Uni- 
versity.   - 


INFLUX  OF  FRENCH  MATHEMATICS.  177 

111  1869  William  E.  Arnold,  major  U.  S.  Volunteers,  entered  upon  the 
duties  of  assistant  i)rofessor  of  mathematics  and  military  tactics,  and 
served  seven  years  in  that  capacity. 

Appointed  as  assistant  professor  at  the  same  time  as  Professor  Arnold, 
was  Henry  T.  Eddy.  He  is  a  native  of  Massachusetts,  was  graduated 
at  Yale  io  1867,  and  then  studied  engineering  at  the  Sheffield  Scientific 
School.  In  1868  he  became  instructor  of  mathematics  and  Latin  at  the 
University  of  Tennessee,  at  Knoxville.  At  Cornell  he  received  the  de- 
grees of  C.  B.  and  Ph.  D.  for  advanced  studies  in  pure  and  applied 
mathematics.  In  1872  he  went  to  Princeton,  where,  for  one  year,  he 
was  associate  professor  of  mathematics.  Since  1874  he  has  held  the 
chair  of  mathematics  at  the  University  of  Cincinnati.  The  year  1879-80 
was  spent  by  him  in  study  abroad. 

Professor  Eddy  has  won  distinction  as  an  original  investigator. 
His  Eesearches  in  Graphical  Statics  (New  York,  1878)  and  his  Neue 
Gonstructionen  in  der  graphischen  Statik  (Leipzig,  1880)  are  contributions 
of  much  value,  and,  we  believe,  the  first  original  work  on  this  subject 
by  an  American  writer.  Professor  Eddy  is  contributing  largely  to'sci- 
entific  and  technical  journals.  In  1874  appeared  his  Analytical  Geom- 
etry. At  the  meeting  of  the  American  Association  for  the  Advancement 
of  Science,  in  Philadelphia  in  1884,  Eddy  was  Vice-President  of  Section 
A,  and  delivered  an  address  on  "  College  Mathematics."  Having  been 
connected  as  student  or  teacher  with  several  higher  institutions  of  learn- 
ing, both  classical  and  scientific,  he  was  able  to  speak  from  his  own  obser- 
vation and  experience  of  the  defects  of  the  mathematical  instruction  in 
the  United  States.    His  address  contains  many  valuable  suggestions. 

In  1870  Lucien  Augustus  Wait,  who  had  just  graduated  at  Harvard, 
was  appointed  assistant  professor.  He  held  this  position  for  about  ten 
years,  when  he  was  made  associate  professor,  which  position  he  still 
holds.  Some  time  ago  he  spent  one  year  in  Europe  on  leave  of  absence. 
Professor  Wait  is  an  energetic  and  excellent  teacher  of  mathematics. 

For  three  years  succeeding  1873  William  E.  Byerly,  a  graduate  of 
Harvard,  and  now  professor  there,  was  assistant  professor  at  Cornell 
University.  Professor  Byerly  is  a  fine  teacher,  and  by  his  publications 
has  made  his  name  widely  known  among  American  ^students  of  the 
more  advanced  mathematics. 

Since  1877  George  William  Jones  has  been  assistant  professor.  He 
is  a  graduate  of  Yale,  1859.  He  "  is  thoroughly  logical,  and  the  best 
drill-master"  in  the  mathematical  faculty  at  Cornell  University. 

As  has  been  seen  from  the  above,  two  of  the  former  assistant  pro- 
fessors at  Cornell  have  since  won  distinction  elsewhere.  The  same  is 
true  of  some  of  the  instructors  in  mathematics.  Before  us  lie  the  names 
of  the  following  former  instructors  in  mathematics  at  Cornell:  George 
Tayloe  Winston  (one  year,  1873,  now  at  the  University  of  North  Caro- 
lina), Edmund  De  Breton  Gardiner  (one  year,  1876),  Charles  Ambrose 
Van  Velzer  (one  year,  1876,  now  professor  of  mathematics  at  University 
881— No.  3—12 


178  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

of  Wisconsin),  Madison  M.  Garver.  and  Morris  E.  Conable  (each  for  part 
of  one  year,  about  187G). 

At  i)resent  there  are  four  instructors,  viz :  James  McMahou  (since 
1884),  Arthur  Staiford  Hathaway  (since  1885),  Duane  Studley  (since 
1887),  George  Egbert  Fisher  (since  1887). 

Mr.  McMahon  is  a  graduate  of  the  University  of  Dublin,  Ireland, 
1881.  He  has  a  fine  mathematical  mind,  and  has  obtained  gold  medals 
for  his  proficiency  in  mathematics  and  mathematical  physics,  and  also 
an  appoiutment  to  a  scholarship  at  his  alma  mater.  He  has  not  pub- 
lished much,  but  has  assisted  in  the  preparation  of  text-books  on  mathe- 
matics issued  by  the  Cornell  professors. 

Mr.  Hathaway  graduated  at  Cornell  in  1879,  and  then  pursued  grad- 
uate studies  at  the  Johus  Hopkins  University,  under  Sylvester  and  his 
associates  till  1884.  While  in  Baltimore  he  frequently  contributed 
papers  to  the  mathematical  society  at  the  university,  which  were  sub- 
sequently published  in  the  Johns  Hopkins  University  Circulars.  He 
has  made  the  theory  of  numbers  his  specialty,  and  has  contributed  sev- 
eral original  articles  on  the  subject  to  the  American  Journal  of  Mathe- 
matics. He  gives  a  new  theory  of  determinately-combining  ideals. 
Mr.  Hathaway  is  not  only  an  able  mathematician,  but  also  an  expert 
stenographer.  When  Sir  William  Thomson,  of  the  University  of  Glas- 
gow, delivered  acourseof  lectures  on  Molecular  Dynamics  at  the  Johns 
Hopkins  University,  in  October,  1884,  Mr.  Hathaway  exercised  his 
"  power  to  seize  on  every  passing  sound."  These  stenographic  notes 
of  Thomson's  lectures  were  printed  by  the  papyrograph  process  and 
published.  At  Cornell,  Hathaway  has  assisted  in  the  preparation  of 
textbooks,  and  is  now,  with  Professor  Jones,  preparing  a  Projective 
Geometry. 

It  will  be  noticed  that  Harvard  University  has  contributed  the  largest 
share  of  mathematical  talent  to  the  faculty  of  Cornell.  Not  only  are 
Byerly  and  Wait  graduates  of  Harvard,  but  also  Oliver,  the  present 
occupant  of  the  mathematical  chair  at  Cornell.  These  three  sat  at  the 
feet  of  that  Gamaliel,  Benjamin  Peirce,  and  caught  the  inspiring  words 
of  their  great  master. 

James  Edward  Oliver  was  born  in  Maine,  in  1829,  and  was  graduated 
at  Harvard  in  1849.  He  had  then  already  displayed  extraordinary 
mathematical  power,  and  was  at  once  appointed  assistant  in  the  office 
of  the  American  Nautical  Almanac,  at  that  time  in  Cambridge.  In  the 
Harvard  catalogues  of  1854  and  1855  we  find  J.  E.  Oliver  and  T.  H. 
Safford  enrolled  as  mathematical  students  in  the  Lawrence  Scientific 
School,  and  taking  advanced  courses  of  mathematics,  such  as  were 
offered  at  that  time  by  no  other  institution  in  the  land.  In  1871  Oliver 
became  assistant  professor  of  mathematics  at  Cornell,  and  two  years 
later  was  given  full  possession  of  the  chair. 

Professor  Oliver  is  an  extraordinary  man,  and  it  is  interesting  to  lis- 
ten to  what  his  former  pupils  have  to  say  of  him.    Says  Prof.  C.  A.  Vai 


INFLUX  OF  FRENCH  MATHEMATICS.  179 

Velzer :  "  He  is  indeed  a  wonderful  man.  If  Professor  Oliver  bad  some 
of  Sylvester's  desire  for  reputation,  he  would  have  been  heard  from  long 
ago,  and  would  have  been  known  all  over  the  world."  Says  Mr.  A.  S. 
Hathaway:  "Professor  Oliver  is  a  rare  genius,  powerful,  able,  but 
without  the  slightest  ambition  to  publish  his  results.  He  works  in 
mathematics  for  the  love  of  it.  I  have  seen  work  of  his  doue  one  or 
two  years  ago.  Practically  the  same  work  appeared  in  the  American 
Journal  of  Mathematics,  written  by  prominent  authors,  that  I  had 
urged  him  to  publish,  and  which  he  had  promised  to  do,  but  which, 
with  his  characteristic  dilatoriness  and  diffidence  in  this  respect,  he 
failed  to  do  until  it  was  too  late.  I  consider  him  fully  equal  in  point 
of  natural  ability  to  Professor  Sylvester,  and  he  is  better  able  than 
Professor  Sylvester,  I  think,  to  acquire  a  knowledge  of  what  others  have 
done.  He  lacks,  however,  the  energy  and  ambition  of  Professor  Syl- 
vester, and  does  not  concentrate  his  powers  on  any  one  subject.  His 
work  is  im-methodical,  and  leads  in  whatever  direction  his  mind  is  bent 
at  the  moment.  The  result  is  that  he  is  a  far  more  amiable  and  con- 
genial person  to  meet  than  Professor  Sylvester.  He  never  obtrudes  self 
upon  you,  and  wherever  you  may  lead  he  will  follow.  Indeed,  his  sim- 
plicity of  character  and  interest  in  everything  that  interests  anybody 
else  is  one  of  his  greatest  charms.  There  are  few  subjects  in  which  he 
does  not  know  more  than  most  people — you  find  it  out  when  you  are 
talking  with  him — but  he  does  not  seem  to  know  it,  at  least  he  never 
obtrudes  it." 

Professor  Wait  is  described  as  a  "  live  energetic  business  manager, 
who  was  appointed  to  the  position  of  associate  professor  to  supplement 
Professor  Oliver's  shortcomings,  and  to  take  care  of  the  practical  man- 
agement of  the  department.  A  better  man  could  not  have  been  chosen 
to  associate  with  Professor  Oliver.  The  latter  finds  in  Professor  Wait  a 
ready  promoter  of  his  ideas  and  plans,  and  one  who  is  capable  of  carrying 
them  out  in  the  smallest  detail,  and  of  taking  charge  of  the  department 
without  troubling  the  chief." 

Professor  Jones  is  a  good  drill-master.  The  bulk  of  the  work  on 
mathematical  text-books  is  done  by  him.  His  style  has  been  adopted 
throughout.  Professsor  Oliver's  style  is  more  classical*  and  polished, 
but  that  of  Professor  Jones  is  more  suitable  for  elementary  text-books. 
In  consequence,  everything  written  by  any  one  else,  has  been  re-shaped 
more  or  less  by  him. 

The  mathematical  faculty  of  Cornell  have  published  several  text- 
books, going  by  the  name  of  "  Oliver,  Wait,  and  Jones's  Mathematics." 
The  works  in  question  are,  a  Treatise  on  Trigonometry,  a  Treatise  on 
Algebra,  and  Logarithmic  Tables.  In  preparation  are  also  a  Drill- 
Book  in  Algebra,  which  will  be  specially  adapted  to  the  work  of  the 
preparatory  schools,  and  a  Treatise  on  Projective  Geometry. 

The  Treatise  on  Trigonometry  has  been  used  successfully  at  Cornell 
for  eight  years,  and  their  Treatise  on  Algebra  for  two  years.     "For  the 


180  TEACHING    AND    HISTORY    OF    MATHEMATICS. 

regular  classes  (in  algebra)  the  more  difficult  parts  have  been  cut  out; 
but  every  year  nearly  all  that  was  omitted  by  them  has  been  taken  up 
by  volunteer  classes  (all  Freshmen)  with  great  satisfaction  and  profit." 
After  eight  years  of  use  the  Trigonometry  has  been  wholly  rewritten. 

The  Treatise  on  Algebra  is  not  a  book  intended  for  beginners,  but 
primarily  for  students  entering  the  Freshman  class  at  Cornell,  and  who 
have  had  extensive  drill  in  elementary  algebra.  Most  of  our  American 
colleges  would  find  the  book  too  difficult  for  use,  on  account  of  deficient 
preparation  on  the  part  of  students  entering. 

If  we  compare  Oliver,  Wait,  and  Jones's  Algebra  with  algebras  used 
in  our  colleges  ten  or  fifteen  years  ago,  we  discover  most  radical  differ- 
ences and  evidences  of  a  speedy  awakening  of  mathematical  life  among 
us.  A  great  shaking  has  taken  place  among  the  "  dry-bones  "  of  Amer- 
ican mathematical  text-books,  and  no  men  "shake"  more  vigorously 
than  the  professors  at  Cornell.  Among  the  improvements  we  would 
mention  a  clearer  statement  of  first  principles  and  of  the  philosophy 
of  the  subject,  the  introduction  of  new  symbols,  a  more  extended  treat- 
ment and  graphic  representation  of  imaginaries,  and  a  more  rigid  treat- 
ment of  infinite  series.  With  some  corrections  and  alterations  in  a  subse- 
quent edition,  we  have  little  doubt  that  the  book  will  become  the  peer  of 
any  algebra  in  the  English  language. 

At  Cornell  great  eflbrts  are  made  to  teach  the  logic  of  mathematics, 
but  it  is  hard  to  attain  the  desired  standard  on  account  of  the  way  that 
preparatory  schools  train  their  pupils.  The  preliminary  training  in 
algebra  generally  gives  students  the  idea  that  algebra  is  merely  a  mass 
of  rules,  and  that  students  have  simply  to  learn  the  art  of  applying  them. 
In  consequence  of  this,  there  is  a  constant  rebellion  among  the  average 
Freshmen  to  the  logical  study  of  algebra.  Formulae  and  substitutions 
are  his  standby. 

The  attendance  of  students  has  been  very  large  at  Cornell.  Com- 
pared with  some  other  departments  of  the  university,  the  teaching 
force  in  mathematics  has  been  rather  small.  In  consequence  of  this, 
the  time  and  energy  of  the  professors  have  been  taxed  unusually  by 
work  in  the  class-room.  In  the  appendix  to  the  Annual  Report  of  the 
President  of  Cornell  University  for  1886-87,  Professor  Oliver  speaks 
of  this  subject,  and  also  of  the  general  work  of  the  mathematical  de- 
partment.    He  says; 

"  We  are  not  unmindful  of  the  fact  that  by  publishing  more,  we  could 
help  to  strengthen  the  university,  and  that  we  ought  to  do  so  if  it  were 
possible.  Indeed,  every  one  of  us  five  is  now  preparing  work  for  pub- 
lication or  expects  to  be  doing  so  this  summer,  but  such  work  progresses 
very  slowly  because  the  more  immediate  duties  of  each  day  leave  us  so 
little  of  that  freshness  without  which  good  theoretical  work  can  not  be 
done. 

"  A  reprint  of  our  algebra,  increased  to  412  pages,  has,  however,  ap- 
peared this  year,  and  has  attracted  favorable  notice  from  the  press  and 


INFLUX  OF  FRENCH  MATHEMATICS.  181 

from  distinguished  matliematicians.  All  five  of  us  have  in  some  way- 
contributed  to  the  work,  but  much  more  of  it  has  been  done  by  Profes- 
sor Jones  than  by  any  one  else.  The  chapters  with  which  we  propose 
to  complete  the  book  deal  mainly  with  special  applications,  or  with 
topics  peculiar  to  modern  analysis.  Meanwhile  we  have  successfully 
used  the  volume  in  all  the  Freshman  sections  this  year.    *    *    * 

"  The  greatest  hindrance  to  the  success  of  the  department,  especially 
in  the  higher  kinds  of  work,  lies,  as  we  think,  in  the  excessive  amount 
of  teaching  required  of  each  teacher ;  commonly  from  seventeen  to 
twenty  or  more  hours  per  week.  The  department  teaches  more  men,  if 
we  take  account  of  the  number  of  hours'  instruction  given  to  each,  than 
does  any  other  department  in  the  university.  Could  each  teacher's 
necessary  work  be  diminished  in  quantity,  we  are  confident  that  the 
difference  would  be  more  than  made  up  in  quality  and  increased  attract- 
iveness." 

From  the  Eeport  for  1887-88,  p.  75,  we  clip  the  following : 

"  Of  course  one  important  means  toward  this  end  [of  securing  the 
attendance  of  graduate  students]  is  the  publication  of  treatises  for  teach- 
ing, and  of  original  work.  A  little  in  both  lines  has  been  done  during 
the  past  year,  though  less  than  would  have  been  but  for  the  pressure 
of  other  university  work,  and  less  than  we  hope  to  accomplish  next 
year.  Professor  Oliver  has  sent  two  or  three  short  articles  to  the  An- 
alyst,* and  has  read,  at  the  National  Academy's  meeting  in  Washing- 
ton, a  preliminary  paper  on  the  Sun's  Eotation,  which  will  appear  in  the 
Astronomical  Journal.  Professor  Jones  and  Mr.  Hathaway  have  lith- 
ographed a  little  Treatise  on  Projective  Geometry.  Mr.  McMahon  has 
sent  to  the  Analyst  a  note  on  circular  points  at  infinity,  and  has  also 
sent  to  the  Educational  Times,  London,  solutions  (with  extensions)  of 
various  problems.  Other  work  by  members  of  the  department  is  likely 
to  appear  during  the  summer,  including  a  new  edition  of  the  Treatise 
on  Trigonometry." 

As  to  the  terms  for  admission  to  the  university,  in  mathematics,  the 
requirements  in  1869  were  arithmetic  and  algebra  to  quadratic  equa- 
tions ;  but  plane  geometry  also  was  required  for  admission  for  the  course 
in  arts.  "  I  j  udge  from  an  old  '  announcement,' "  says  Professor  Oliver, 
"  that  in  1868,  when  the  university  opened,  some  students  were  ad- 
mitted with  only  arithmetic."  In  recent  years  the  requirements  have 
been  arithmetic,  algebra  through  quadratics^  radicals,  theory  of  expo- 
nents, and  plain  geometry.  In  the  engineering  and  architectural  courses 
solid  geometry  has  been  added. 

In  and  after  1889,  candidates  will  have  two  examinations,  the  "  pri- 
mary "and  the  "advanced."  The  "primary"  examination  will  cover 
the  following  subjects  in  mathematics  : 

In  Arithmetic,  including  the  metric  system  of  weights  and  measures;  as  much  as 
is  contained  in  the  larger  text-books. 

*The  name  of  the  mathematical  journal  in  question  is  not  Analyst,  but  Annals  of 
Mathematics, 


182  TEACHING   AND   HISTORY   OF  MATHEMATICS. 

In  Flane  Geoineiry  ;  as  mucli  as  is  contained  in  the  firsfc  five  books  of  Chauvenet's 
Treatise  on  Elementary  Geometry,  or  in  the  first  five  books  of  Wentworth's  Elements 
of  Plane  and  Solid  Geometry,  or  in  the  first  six  books  of  Newcomb's  Elements  of  Ge- 
ometry, or  in  the  first  six  books  of  Hamblin  Smith's  Elements  of  Geometry. 

In  Algebra,  through  quadratic  equations,  and  including  radicals  and  the  theory  of 
exponents;  as  much  as  is  contained  in  the  corresponding  parts  of  the  larger  treatises 
of  Newcomb,  Oluey,  Ray,  Eobinson,  Todhunter,  Wells,  or  Wentworth,  or  m  those 
parts  of  Oliver,  Wait,  and  Jones's  Treatise  on  Algebra  that  are  indicated  below,  with, 
the  corresponding  examples  at  the  ends  of  the  several  chax^ters  :  Chapters  I,  II,  III ; 
Chapter  IV,  except  theorems  4,  5,  6 ;  Chapter  V,  except  ?§  3,  5,  and  notes  3,  4,  of 
problem  2;  Chapter  VII,  §  U  ;  Chapter  VIII,  ^  1,  2,  the  first  three  pages  of  §  8  and 
$  9 ;  Chapter  XI,  except  §  9,  problem  9  of  §  12,  and  §§  13,  17,  18. 

For  admission  to  the  course  leading  to  tbe  degree  of  bachelor  of  arts, 
no  further  knowledge  of  mathematics  will  be  necessary,  in  any  case. 

For  admission  to  the  courses  leading  to  the  degrees  of  bachelor  of 
philosophy,  bachelor  of  science,  bachelor  of  letters;  to  the  course  in 
agriculture )  and  (in  and  after  1890)  for  all  optional  students,  there  will 
be  required,  in  addition  to  the  "  primary  "  examination,  an  "  advanced  " 
examination  in  two  advanced  subjects,  "one  of  which  must  be  French 
or  German  or  mathematics."  If  the  applicant  chooses  mathematics,  he 
will  be  examined  on  all  the  Freshman  mathematics,  namely,  solid  geom- 
etry and  elementary  conic  sections,  as  much  as  is  contained  in  New- 
comb's  Elements  of  Geometry  ;  advanced  algebra,  as  much  as  is  con- 
tained in  those  parts  of  Oliver,  Wait,  and  Jones's  Treatise  on  Algebra 
that  are  read  at  the  university  (a  list  is  sent  on  application  to  the  Reg. 
istrar) ;  and  trigonometry,  plane  and  spherical,  as  much  as  is  contained 
in  the  unstarred  portions  of  Oliver,  Wait,  and  Jones's  Treatise  on 
Trigonometry. 

It  was  the  desire  of  Mr.  Cornell  and  President  White  to  establish  a 
university  giving  broad  and  general  training,  in  distinction  to  the  nar- 
row, old-fashioned  college  course  with  a  single  combination  of  studies. 
The  idea  was  well  expressed  by  Cornell  when  he  said  that  he  trusted 
the  foundation  ha^  been  laid  to  "  an  institution  where  any  person  can 
find  instruction  in  any  study."  We  shall  proceed  to  give  the  course  of 
study  in  mathematics,  and  let  the  reader  judge  for  himself  whether  or 
not  the  idea  of  the  founder  has  been  carried  out  in  the  mathematical 
department. 

We  begin  with  studies  which  have  been  regmVe^  for  graduation.  The 
mathematical  course  has  always  included,  for  all  candidates  for  bacca- 
laureate degrees  except  (at  one  time)  a  few  natural  history  and  analytic 
chemistry  students,  one  term  each  of  solid  geometry,  advanced  algebra, 
and  trigonometry  (either  plane,  or  plane  and  spherical).  At  one  time 
students  in  history  and  political  science  had  one  term  of  theory  of  prob- 
abilities and  statics  instead  of  spherical  trigonometry.  There  have  also 
always  been  required  in  all  engineering  courses  and  in  architecture, 
analytic  geometry  and  calculus ;  and,  sometimes,  analytic  geometry  in 
certain  other  courses,  as  those  in  science  and  philosophy.  At  present 
the  amount  required  is  one  term  of  analytic  geometry  and  one  term  of 


INFLUX  OF  FRENCH  MATHEMATICS.  183 

calculus,  in  the  course  of  architecturCj  and  one  term  of  analytic  geom- 
etry and  two  terms  of  calculus  in  tbe  engineering  course.  At  pres- 
ent, the  students  in  mechanical  and  electric  engineering  take  also  an 
extra  term  in  projective  geometry  and  theory  of  equations  in  the  Fresh- 
man year.  These  are  the  "required"  mathematics  in  the  different 
courses. 

In  addition  to  these, "  elective  "  mathematics  has  always  been  ofi'ered 
by  the  university  to  upper  classmen,  and  also,  of  late  years,  to  Freshmen 
and  Sophomores.  The  number  of  these  elective  courses  has  gradually 
increased,  till  now  they  are  as  follows  (Register  1888-89) : 

ELECTIVE  WORK.* 

[Any  course  not  desired  at  the  beginning  of  the  fall  term  by  at  least  three  students, 
properly  prepared,  may  not  be  given.] 

11.  Problems  in  Geometry,  Algebra  and  Trigonometry,  supplementary  to  the  pre- 
scribed work  in  those  subjects,  two  hours  a  week.    Professor  Jones. 

12.  Advanced  work  in  Algebra,  including  Determinants  and  the  Theory  of  Equa- 
tions, two  hours  a  week.     Professor  Wait. 

13.  Advanced  work  in  Trigonometry,  one  hour  a  week.     Professor  Wait. 

[The  equivalents  of  courses  8,  12,  and  13  are  necessary,  and  course  11  is  useful,  as 
a  preparation  for  most  of  the  courses  that  follow.] 

14.  Advanced  work  in  Analytic  Geometry  of  two  and  three  Dimensions,  viz  : — 

(a)  First  year,  Lines  and  Surfaces  of  First  and  Second  Orders.  3  hours.  Professor 
Jones. 

(6)  Second  year.  General  Theory  of  Algebraic  Curves  and  Surfaces.  3  hours.  Pro- 
fessor Oliver. 

15.  Modern  Synthetic  Geometry,  including  Projective  Geometry.  2  hours.  Pro- 
fessor Jones. 

16.  Descriptive  and  Physical  Astronomy.     3  hours.     Mr.  Studley. 

17.  The  Teaching  of  Mathematics.  Seminary  work.  1  hour.  Professor  OLIVER, 
and  most  of  the  teachers  in  the  Department. 

18.  (a)  Mathematical  Essays  and  Theses:  (&)  Seminary  for  discussion  of  results  of 
students'  investigations.     Professor  Oliver. 

19.  Advanced  work  in  Differential  and  Integral  Calculus.     3  hours.    Mr.  Fisher. 

20.  Quantics,  with  Applications  to  Geometry.  Eequires  courses  8,  12,  14  (a),  and 
preferably  also  11,  13,  19.  May  be  simultaneous  with  14  (&).  3  hours.  Mr.  Mc- 
Mahon. 

21.  Dififerential  Equations  :  to  follow  course  19.    3  hours,    Mr.  Hathaway. 

22.  Theory  of  Functions.  Requires  course  19,  and  preferably  21.  (a)  First  year, 
3  hours.     (6)  Second  year,  2  hours.     Professor  Oliver. 

23.  Celestial  Mechanics.     3  hours.     Professor  Oliver. 
25.  Finite  Differences.     2  hours.     Professor  Oliver. 

27.  Rational  Dynamics.     Professor  AVait. 

28.  Molecular  Dynamics;  or,  29,  Theory  of  Numbers.     3  hours.     Mr.  Hathaway. 

30.  (o)  Vector  Analysis;  or,  (&)  Hyper-Geometry;  or,  (c)  Matrices  and  Multiple 
Algebra.     2  hours.     Professor  Oliver. 

31.  Theory  of  Probabilities  and  of  Distribution  of  Errors,  including  some  sociologic 
applications.     2  hours.    Professor  Oliver,  or  Professor  Jones. 

41.  Mathematical  Optics,  including  Wave  Theory  and  Geometric  Optics.  2  hours. 
Professor  Oliver. 

*  Numbers  1  to  10,  inclusive,  refer  in  the  catalogue  to  required  studies  in  mathe- 
matics. 


184  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

43.  Mathematical  Theory  of  Sound.     3  hours.     Mr.  McMahox* 

44.  Mathematical  Theory  of  Electricity  and  Magnetism.  Professors  Oliver  and 
Wait. 

In  most  of  the  aboye  branches  of  j)ure  mathematics,  an  additional  year's  instruc- 
tion, 1  or  2  hours  per  week,  may  be  given  if  desired. 

For  several  years  (from  1874  to  1887,  we  believe)  tbere  has  been  also 
a  "course  in  mathematics,"  with  a  fised  curriculum,  leading  to  the 
degree  of  "bachelor  of  science  in  mathematics,"  but  it  was  dropped  when 
the  numerous  prescribed  curricula  and  resulting  degrees  were  consoli- 
dated into  a  few  "  general  courses,"  of  which  the  work  is  mainly  pre- 
scribed in  the  first  two  years  and  mainly  elective  in  the  last  two,  and  a 
few  "  technical  courses,"  whose  work  is  mainly  prescribed  throughout. 
That  old  "course  in  mathematics"  comprised  some  language  and  culture 
studies,  botany,  geology,  logic,  English  literature,  descriptive  geom- 
etry, analytical  mechanics,  lectures  and  laboratory  work  in  physics, 
while,  perhaps,  two-fifths  of  all  the  student's  time  was  given  to  pure 
mathematics,  including  analytical  geometry,  calculus,  differential  equa- 
tions, finite  differences,  quaternions,  imaginaries,  mathematical  essays, 
seminary- work,  etc.  The  object  of  this  course  was  to  give  the  best 
equipment  to  students  intending  to  become  teachers  of  mathematics, 
professors,  and  investigators.  The  students  in  this  course  were  few, 
but  earnest,  and  some  of  them  have  since  been  making  their  mark  as 
teachers  and  investigators. 

As  to  the  mathematical  text-books  which  have  been  used  at  different 
times,  we  make  the  following  statement : 

In  Elementary  Geometrj^,  Loomis  till  about  1873 ;  since  then,  Chau- 
venet. 

In  Elementary  Geometric  Conies,  Loomis,  thenPeck,  though  the  pres- 
ent professors  "  don't  much  like  either." 

In  Modern  Synthetic  Geometry,  Professor  Evans  used  no  book,  but 
gave  lectures.  The  same  has  sometimes  been  done  since.  At  other 
times,  Cremona's  Geometrie  ProjecUvej  or  the  recent  English  translation, 
was  used.  But  now  a  little  lithographed  treatise  on  Projective  Geome- 
try, written  for  the  puri^ose  by  Jones  and  Hathaway,  is  being  used. 
Professor  Oliver  has  taught,  also,  Casey's  Sequel  to  Euclid,  and,  once, 
Steiner's  Conies. 

In  Algebra,  first  Loomis,  then  Davies'  Bourdon,  Olney,  Wells,  New- 
comb,  and  now  Oliver,  Wait,  and  Jones's,  Todhunter's,  Burnside  and 
Panton's  Theory  of  Equations. 

In  Determinants,  Muir,  Dostor^  Hanus,  and  lectures. 

In  Quautics,  Salmon's  Higher  Algebra. 

In  Trigonometry,  first  Loomis's  (including  a  little  of  mensuration, 
.surveying,  and  navigation),  then  Greenleaf's,  Chauvenet's,  Wheeler's, 
and  now  Oliver,  Wait,  and  Jones's,  Todhunter's. 

In  Analytic  Geometry,  first  Loomis  (for  two  dimtmsions)  and  Davies 
(for  three  dimensions),  also  Church,  then  Peck,  Todhuuter,  Aldis,  and 
now  Smith  (English  work)  with  the  three  dimensioJas  by  lecture.  With 


INFLUX  OF  FRENCH  MATHEMATICS.  185 

more  advanced  students  have  been  used  also  Salmon's  Conic  Sections, 
Higher  Plane  Curves,  and  Analytic  Geometry  of  Three  Dimensions. 

In  Differential  and  Integral  Calculns,  first  Loomis  and  Church,  then 
Peck,  Todhunter,  Williamson,  Taylor,  Meunier-Joannet,  Homersham 
Cox,  Woolhouse,  Smyth,  Byerly ;  and  now  Taylor  for  the  few  students 
in  the  one-term  course,  the  abridged  Kice  &  Johnson's  Differential  and 
Eice's  Integral  Calculus  (one  term  each)  for  the  two-term  course  for 
engineers,  and  Williamson  and  Todhunter  for  advanced  work,  with 
Bertrand  for  occasional  reference  and  special  work. 

In  Imaginaries,  Argand  was  used,  but  now  preference  is  given  to 
Chapter  X  of  Oliver,  Wait,  and  Jones's  Algebra, 

In  EquipoUences,  Belavitis  was  once  used. 

In  Quaternions,  Kelland,  Tait,  Hardy,  Hamilton's  Lectures,  Hamil- 
ton's Elements. 

In  Theory  of  Functions  there  have  been  used  Laurent's  Fonctions 
Mliptiques,  Hermite's  Cours  cV  Analyse  ;  and  now  Briot  and  Bouquet's 
Theorie  des  Fonctions  Flliptiques  and  Halphen's  Traite  des  Fonctions 
Flliptiques. 

In  Theory  of  Numbers,  Dedekind's  edition  of  Lejeune  Dirichlet's 
Zahlenthcorie  has  been  used  recently. 

In  Least  Squares,  Merriman. 

In  Differential  Equations,  Boole,  Forsyth. 

In  Finite  Differences,  Boole. 

In  Descriptive  Astronomy,  Loomis,  Newcomb,  and  Holden,  with 
Young's  "The  Sun"  and  Chauvenet  (for  eclipses)  for  collateral  reading. 

In  Mechanics,  Duhamel's  Mecanique  Analytique,  and  now  Minchin's 
Analytical  Statics  and  Williamson's  Analytical  Dynamics. 

Quaternions  have  not  been  taught  now  for  several  years,  because  the 
professors  are  convinced  that  the  benefit  of  that  study  is  Avith  most 
students  better  gotten  with  a  mixed  course  in  matrices,  vector  addition 
and  subtraction,  imaginaries^  and  theory  of  functions. 

Among  the  fundamental  ideas  of  President  White,  in  organizing  the 
university,  was  a  close  union  of  liberal  and  practical  education.  There 
have,  therefore,  from  the  beginning,  existed  separate  departments  of 
civil  engineering,  of  mechanic  arts,  and  of  physics,  each  with  a  sepa- 
rate professor  at  its  head.  Astronomy  is  taught  partly  in  the  depart- 
ment of  civil  engineering  and  partly  in  that  of  mathematics. 

Pupils  in  mathematics  are  always  encouraged  to  do  original  work, 
but  it  is  only  by  older  and  maturer  students  that  researches  are  made 
which  are  of  sufficient  value  to  merit  publication.  The  writer  has 
before  him  two  printed  theses,  written  to  secure  the  degree  of  doctor  of 
philosophy  at  Cornell  University.  One  is  by  C.  E.  Linthicum,  "On  the 
Eectification  of  Certain  Curves,  and  on  Certain  Series  Involved"  (Balti- 
more, 1888) ;  the  other  is  by  RoMin  A.  Harris,  on  "The  Theory  of  Im- 
ages in  the  Representation  of  Functions"  (Annals  of  Mathemetics, 
June,  1888).  Both  of  these  are  very  creditable  to  the  writers  and  to 
the  university,  and  the  latter  appears  to  us  to  fill  a  gap. 


186  TEACHING   AND   HISTORY    OF    MATHEMATICS. 

There  are  always  some  imder-graduate  students  who  do  good  work 
in  the  more  advanced  mathematical  electives,  but  at  present  it  is  by 
resident  graduates  of  Cornell  and  other  colleges  that  the  best  advanced 
work  is  expected  to  be  done.  Great  efforts  have  been  and  are  being 
made  to  secure  the  attendance  of  graduate  students  in  advanced  courses 
in  mathematics.  During  the  year  1885-86  eleven  graduate  students 
were  engaged  in  the  study  of  the  higher  mathematics.  The  number  for 
the  year  following  is  not  known  to  the  writer,  but  the  president's  re- 
port indicates  that  the  attendance  on  advanced  courses  in  mathematics 
was  increasing,  and  that  about  one-fifth  of  the  graduate  students  were 
taking  their  chief  work  in  mathematics.  In  the  last  report  Professor 
Oliver  says : 

"  During  1887-88  eleven  graduate  students  have  taken  more  or  less 
of  their  work  with  us.  Allowing  for  such  as  were  partly  in  other  de- 
partments or  remained  but  part  of  the  year,  we  find  that  the  mathe- 
matical department  has  had  about  one-seventh  of  all  the  graduate  work 
in  the  university.  This  would  seem  to  be  our  full  share  of  this  desirable 
kind  of  teaching,  when  it  is  considered  that  the  higher  mathematics  is 
diflScult,  abstract,  and  hard  to  popularize;  that  of  course  we  can  not 
attract  students  to  it  by  laboratories  and  large  collections  (except  of 
books),  nor  by  the  prospect  of  lucrative  industrial  applications ',  and 
that  our  department's  whole  teaching  force,  composed  of  only  about 
one-eleventh  of  all  the  active  resident  professors  and  instructors  in  the 
university,  and  including  only  one-thirteenth  of  the  resident  professors, 
has  to  do  about  one-ninth  of  all  the  teaching  in  the  university." 

We  are  sure  that  many,  perhaps  all  of  our  professors  of  mathematics 
will  see  in  the  following  remarks  by  Professor  Oliver  the  reflection  of 
their  own  experience  as  teachers  : 

"We  have  always  had  to  contend  with  one  other  serious  difficulty. 
There  is  a  wide-spread  notion  that  mathematics  is  mainly  important  for 
the  preliminary  training  of  certain  crude  powers,  and  as  auxiliary  to 
certain  bread-winning  professions,  and  that  only  literary  studies  can 
afford  that  fine  culture  which  the  best  minds  seek  for  its  own  sake. 
Time,  no  doubt,  will  rectify  this  misapprehension ;  but  meanwhile  it 
hinders  our  success." 

The  methods  of  teaching  mathematics  at  Cornell  are  various.  The  pro- 
fessors sometimes  lecture,  especially  when  there  is  no  suitable  textbook 
at  hand.  This  method,  when  a  rather  full  syllabus  is  given  out  before- 
hand, and  plenty  of  problems  are  assigned  to  the  students  for  solution, 
has  sometimes  proved  very  successful.  The  lecturer  perhaps  calls  upon 
the  class  for  suggestions  as  he  proceeds  with  his  topic,  and  then  assigns 
to  them  for  home  study  some  problem  very  much  alike  in  principle  to 
the  one  they  have  just  been  discussing  together. 

Butoftener  it  is  preferred  to  base  the  teaching  upon  a  book  that  the  stu- 
dents can  study  for  themselves,  supplementing  it  by  lectures  and  expla- 
nations, and  holding  the  class  to  recitations  and  examinations  upon  it, 
In  all  the  work,  and  especially  in  that  for  advanced  classes,  the  pupils 


INFLUX  OF  FRENCH  MATHEMATICS.  187 

'  are  treated  by  tlie  professor  as  fellow- students,  and  he  avoids  assuming 
toward  them  the  air  of  master  and  dictator.  Independent  thought  is 
constantly  encouraged,  even  when  this  leads  the  students  to  criticise 
the  things  they  are  being'  taught.  Mere  memory-work  and  rote-learn- 
ing—still in  vogue  in  many  of  our  schools— is  discouraged  in  every  way 
possible. 

Some  of  the  mathematical  teachers  at  Oornell  have  been  accustomed 
to  test  their  pupil's  mastery  of  the  subject  by  written  examinations, 
given  in  the  midst  of  the  term's  work  without  warning,  or  on  weekly 
reviews.  There  is  also  a  written  examination  at  the  close  of  each  term ; 
but  students  who  have  done  their  term's  work  with  a  certain  degree  of 
excellence  beyond  what  would  be  strictly  requisite  to  "  pass  them  up" 
in  the  subject  are  often  exempt  from  this  examination. 

Since  1874  mathematical  clubs  have  existed  at  times.  Different 
members  of  it  would  give  in  turn  the  results  of  their  mathematical 
studies  in  lines  a  little  outside  of  the  regular  work  of  the  class-room, 
and  the  matter  thus  presented  was  then  open  to  discussion  by  the  whole 
company  present.  Professor  Oliver  has  generally  presided  at  these 
meetings  and  taken  his  turn  at  presenting  topics  and  work  for  discus- 
sion. The  attendance  upon  these  clubs  has  generally  been  small,  in- 
cluding only  the  professors,  instructors,  and  a  few  advanced  students. 
Sometimes  the  meetings  would  be  kept  up  for  a  few  months  or  a  year 
with  a  good  deal  of  spirit,  and  then  with  change  of  membership  the  in- 
terest would  flag,  and  the  club  would  be  discontinued  for  a  while. 
Much  of  the  work  presented  was  the  work  of  immature  students  and 
not  worth  publishing.  But  these  clubs  have  helped  to  keep  up  an  in- 
terest in  mathematics  and  to  stimulate  the  spirit  of  originality. 

For  the  past  three  or  four  years  the  club  has  been  merged  into  a 
"  seminary"  for  the  discussion  of  aims  and  methods  in  teaching  math- 
ematics. Here  the  professor  proposes  such  problems  as  these:  "Why 
do  we  teach  mathematics  at  all,  and  what  practical  rules  does  this  sug- 
gest to  us  in  order  that  our  teaching  may  be  most  effective  and  useful 
toward  the  end  i)roposed  f  "  What  is  the  place  of  memory  in  math- 
ematical teaching?"  "What  are  the  relative  advantages  of  lecturing 
and  book  work,  and  how  are  they  best  combiuedf  "  How  can  we  best 
teach  geometryf  "What  is  the  nature  of  axioms  in  geometry,  and 
how  modified  when  we  consider  the  possibility  of  non  Euclidian  space '?" 
The  professor  proposes  some  such  problem,  then  calls  for  discussion  and 
adds  his  own  views.  If  possible  he  develops  on  the  blackboard  a  sylla- 
bus or  tabular  view  of  the  different  heads  under  which  the  theory  must 
fall.  Then  these  are  discussed  in  order,  either  at  that  or  at  subsequent 
meetings.  In  the  latter  case  the  discussions  are  often  opened  by  es- 
says from  members  of  the  seminary.  This  method  of  conducting  the 
seminary  is  most  fruitful  of  results,  especially  if  we  remember  that  the 
chief  object  of  the  graduate  department  in  mathematics  is  to  train 
teachers  of  this  science.  The  coming  teacher  will  acquire  possession  of 
better  methods  and  higher  ideals  of  mathematical  teaching. 


188  TEACHING   AND   HISTORY   OP  MATHEMATICS, 

VIKaiNIA  MILITARY  INSTITUTE. 

The  Virginia  Military  Institute  at  Lexington,  Ya.,  is  a  State  institu- 
tion, and  was  organized  in  1839  as  a  military  and  scientific  school.  It 
is  a  foster  child  of  the  U.  S.  Military  Academy  at  West  Point.  At  its 
organization  General  Francis  H.  Smith  was  made  its  superintendent. 
This  position  he  has  now  held  for  half  a  century.  What  the  Yirginia 
Military  Institute  has  been  and  is,  is  due  chiefly  to  his  long  and  faithful 
service  as  superintendent. 

General  Francis  Henney  Smith  is  a  native  of  Virginia.  He  graduated 
at  West  Point  in  1833,  and  was  assistant  professor  of  mathematics  there 
during  the  first  two  years  after  graduation.  He  then  occupied  the 
chair  of  mathematics  for  two  years  at  Hampden-Sidney  College.  At 
the  military  institute  he  added  to  his  duties  as  superintendent  those  of 
professor  of  mathematics  and  moral  philosophy. 

Smith  has  published  a  number  of  mathematical  text-books.  Some  of 
his  books  have  suffered  from  frequent  typographical  errors.  In  1845 
appeared  his  American  Statistical  Arithmetic,  in  the  jjreparation  of 
which  he  was  aided  by  E.  T.  W.  Duke,  assistant  professor  of  mathe- 
matics at  the  institute.  The  book  was  called  "  Statistical  Arithmetic," 
because  the  examples  were  selected  as  far  as  practicable  from  the  most 
prominent  facts  connected  with  the  history,  geography,  and  statistics 
of  our  country.  This  novel  idea  made  that  arithmetic  the  medium  for 
communicating  much  important  information  and  a  better  appreciation 
of  the  greatness  and  resources  of  our  country. 

Other  arithmetics  appeared  by  the  same  author,  which  enjoyed  quite 
an-  extensive  circulation.  About  1848  was  published  also  a  series  of 
algebras,  as  a  part  of  the  mathematical  series  of  the  Virginia  Military 
Institute. 

A  valuable  contribution  to  the  list  of  college  text-books  was  the 
translation,  by  Professor  Smith,  in  1840,  of  Biot's  Analytical  Geometry. 
The  original  French  work  of  Biot  was  for  many  years  the  text-book  for 
the  U.  S.  Military  Academy  at  West  Point.  When,  about  ten  years 
previous,  Professor  Farrar  prepared  his  Cambridge  mathematics,  he 
cliose  Bezout's  work  on  the  "  application  of  algebra  to  geometry,"  in 
preference  to  the  works  of  Lacroix  and  Biot,  for  the  reason  that  these 
works  were  thought  to  be  too  advanced  for  our  American  colleges, 
which  had  up  to  that  time  paid  no  attention  whatever  to  analytical 
geometry.  Bezout's  work  can  hardly  be  called  an  analytical  geometry. 
The  only  works  on  this  subject  which  were  published  in  this  country 
after  the  Cambridge  mathematics  and  previous  to  Smith's  Biot  were 
the  elementary  treatise  of  J.  E.  Young  (which  followed  Bourdon  as  a 
model)  and  the  work  from  the  pen  of  Professor  Davies,  of  West  Point. 
Smith's  translation  of  Biot  reached  a  second  edition  in  1846.  After- 
ward the  book  was  revised.    An  edition  of  it  appeared  in  1870. 

In  1867  Smith  published  an  edition  of  Legendre's  Geometry.    Edi- 


INFLUX  OF  FRENCH  MATHEMATICS.  189 

tions  of  this  work  had  appeared  in  this  country  by  Farrar  and  Davies. 
Smith's  translation  was  from  a  later  French  edition,  which  contained 
additions  and  modifications  by  M.  A.  Blanchet,  an  61eve  of  the  jScole 
Polytechnique. 

In  1868  appeared  from  the  pen  of  General  Smith  a  Descriptive  Geom- 
etry. The  study  of  this  subject  had  been  introduced  in  the  institute  at 
a  time  when  it  was  hardly  known  by  name  in  other  schools  and  colleges 
of  Yirgiriia. 

The  organization  of  the  Virginia  Military  Institute  and  the  methods 
of  teaching  have  been  much  the  same  as  at  the  U.  S.  Military  Academy. 
Indeed,  the  institute  is  frequently  called  the  "  Southern  West  Point." 
The  division  of  classes  into  sections  and  the  rigid  and  extended  appli- 
cation of  the  '^  marking  system  "  have  been  adopted  from  West  Point. 
The  marking  system  seems  to  have  originated  in  France,  and  to  have 
been  introduced  into  this  country  by  West  Point. 

The  relative  weight  given  to  the  different  subjects  of  instruction 
forming  the  general  merit-roll  of  each  class  is,  according  to  the  Official 
Eegister  of  1887-88,  as  follows : 

1.  Mathematics  (grade).. 3      f  12.  Surveying 1 

•2.  Civil  engineering 3       \  13.  Moral  and  political  philosophy  ..   1 


3.  Military  engineering 1 

4.  Chemistry 2 

5.  Mechanics 2 

6.  French  . 1 

7.  German 1.5 

8.  English 1 

9.  Physics 1.5 

10.  Mineralogy 1 

11.  Astronomy 1 


14.  Ordnance  and  gunnery 1 

15.  Drawing 1 

16.  Geography ,...  1 

17.  Infantry  tactics 0.5 

18.  Geology... 0.5 

19.  Descriptive  geometry 1 

20.  Logic 0.5 

21.  Ehetoric  0.5 

22.  Latin i.5 


The  success  of  the  educational  work  of  the  school  turns  largely  upon 
the  method  of  dividing  classes  into  sections,  whereby  the  students  are 
acGUTSbtolj  graded  according  to  scholarship,  and  each  secures  a  propor- 
tionately large  share  of  the  personal  attention  of  the  instructor.  Each 
section  is  "  under  the  command  of  a  '  section-marcher,'  taken  from  the 
first  cadet  on  the  section-roll.  The  sections  are  formed  on  parade,  at 
the  appointed  hours ;  the  roll  is  called  by  the  section-marcher,  absen- 
tees are  reported  to  the  officer  of  the  day,  whose  duty  it  is  to  order  all 
not  properly  excused  to  the  class  duty.  The  section-marcher  then 
marches  his  section  to  the  class-room,  reports  the  absentees  to  the  pro- 
fessor, and  then  transfers  to  him  the  responsibility  which  he  had  thus 
far  borne.  The  professor  examines  the  section  on  the  appointed  lesson, 
is  responsible  for  the  efficiency  of  hisinstruction,  and  once  a  week  makes 
an  official  report  to  the  superintendent  of  the  progress  of  his  section. 
These  reports  are  duly  recorded,  and  constitute  an  important  element 
in  the  standing  of  each  cadet  at  his  semi-annual  or  general  examina- 
tions."* 

*  The  Inner  Life  of  the  Virginia  Military  Institute  Cadet,  by  Francis  H.  Smith, 
LL.  D.,  1878,  p.  22. 


190  TEACHING   AND    HISTOEY    OP   MATHEMATICS. 

As  at  West  Point,  so  at  this  institution,  a  candidate  for  admission  lias 
been  required  to  know  no  other  mathematical  study  than  arithmetic. 
"  The  four  ground  rules  of  arithmetic,  vulgar  and  decimal  fractions,  and 
the  rule  of  three  "  admitted  a  candidate,  as  far  as  mathematics  is  con- 
cerned. 

The  course  of  study  has  been  the  same  as  at  West  Point,  but  the 
books  used  have  not  always  been  the  same.  The  books  used  at  the 
beginning  were  as  follows  :  Bourdon's  Algebra,  Legeudre's  Geometry, 
Boucharlat's  Analytical  Geometry  (in  French),  Boucharlat's  Differential 
and  Integral  Calculus  (in  French),  Davies'  Descriptive  Geometry. 
These  were,  later,  displaced  by  other  books,  chiefly  Smith's  own  works, 
viz,  Smith's  Algebra,  Smith's  Descriptive  Geometry  (after  De  Fourcy), 
Smith's  Legeudre's  Geometry,  Smith's  Biot's  Analytical  Geometry, 
Oourtenay's  Differential  and  Integral  Calculus,  Buckingham's  Calculus. 
As  at  West  Point,  so  here,  there  have  been  no  elective  studies. 

During  the  first  twenty  years  of  its  existence  the  Virginia  Military 
Institute  was  flourishing.  It  "had  just  placed  itself  before  the  public 
as  a  general  school  of  applied  science  for  the  development  of  agricultural, 
mineral,  commercial,  manufacturing,  and  internal  improvement  interests 
of  the  State  and  country  when  the  army  of  General  Hunter  destroyed  its 
stately  buildings  and  consigned  to  the  flames  its  library  of  ten  thousand 
volumes,  the  philosophical  apparatus  used  for  ten  years  by  'Stonewall' 
Jackson,  and  all  its  chemicals.  The  cadets  were  then  transferred  to 
Richmond,  and  the  institution  was  continued  in  vigorous  operation 
until  the  evacuation  of  Richmond  on  the  3d  of  April,  1865."* 

The  War  left  sad  traces  on  the  institution,  besides  the  destruction  of 
its  buildings,  library,  and  apparatus.  Three  of  its  professors  had  been 
slain  in  battle :  Stonewall  Jackson,  who  had  been  professor  of  natural 
and  experimental  philosophy  since  1850  5  Maj.  Gen.  R.  E.  Kodes,  a 
graduate  of  the  institute,  and,  in  1860,  appointed  professor  of  civil  and 
military  engineering  ;  Col.  S.  Crutchfield,  also  a  graduate  of  the  insti- 
tute, and,  since  1858,  professor  of  mathematics.  Among  the  slain  were 
also  two  assistant  professors  and  two  hundred  of  its  alumni. 

Notwithstanding  the  impoverishment  of  the  people  immediatelj'  after 
the  War,  it  was  decided  in  1865  to  re-open  the  institution.  Without  one 
dollar  at  command  to  offer  by  way  of  salary  to  the  professors,  the  board 
of  visitors  called  back  all  who  survived,  and  filled  the  vacancies  of  those 
who  had  died.  Work  was  begun  with  earnestness.  On  the  18th  of  Oc- 
tober, 1865,  the  day  designated  for  the  resumption  of  academic  duties, 
sixteen  cadets  responded.  At  the  end  of  the  academic  year  the  num- 
ber of  cadets  was  55.  Such  vitality  under  such  discouragements 
prompted  the  legislature  to  restore  the  annuity  the  next  winter.  It  was 
not  very  long  before  the  Virginia  Military  Institute  was  restored  to  all 
its  former  lustre.  In  1870  the  buildings  of  the  institute  were  restored 
and  equipped  with  laboratories  and  instruments. 

*  Official  Eegister,  1887-88. 


INFLUX  OF  FRENCH  MATHEMATICS.  191 

The  OfiQcial  Register  for  1887-88  gives  119  cadets  in  the  academic  school. 
The  studies  in  mathematics  for  that  year  are  as  follows  :  Fourth  class — 
First  year:  Smith's  Algebra,  Davies'  Legendre's  Geometry  and  Trigo- 
nometry (revised  by  Yan  Amringe),  Exercises.  (Recitations  from  8  to 
11  daily.)  Third  class— Second  year:  Smith's  Biot's  Analytical  Geome- 
try, Buckingham's  Diiierential  and  Integral  Calculus.  (Recitations  from 
9  to  11  daily.)  Second  class — Third  year :  Mahau- Wheeler,  Davies'  Sur- 
veying (Van  Amringe),  Gillespie's  Surveying,  field  work.  (Recitation 
from  10  to  11.)  First  class — Fourth  year;  Rankine's  Applied  Mechanics 
and  Rankine's  Civil  Engineering,  lectures,  and  field  practice. 

UNIVERSITY  OF  VIRGINIA. 

President  Jefferson  devoted  the  golden  evening  of  his  life  to  the 
founding  and  building  up  of  the  University  of  Virginia  as  a  nursery  for 
the  youth  of  his  much-loved  State.  This  greatest  university  of  the 
South  has  from  its  beginning  had  features  peculiar  to  itself.  The  entire 
abandonment  of  the  class  system,  and  the  course  arrangement  of  its 
studies,  are  its  most  i)rominent  distinguishing  features.  From  the  very 
beginning  the  method  of  instruction  has  been  by  lectures  and  examina- 
tions. "  Test-books  are  by  no  means  discarded,  but  the  professor  is 
expected  to  enlarge,  explain,  and  supplement  the  text.  Every  lecture 
is  preceded  by  an  oral  examination  of  the  class  on  the  preceding  lecture 
and  the  corresponding  text.  This  method  stimulates  the  professor  to 
greater  efforts,  and  excites  and  maintains  the  interest  and  attention  of 
the  students  a  hundred  fold."* 

The  university  was  opened  for  students  in  March,  1825.  It  then  had 
eight  distinct  schools,  but  at  the  present  time  it  has  nineteen,  "  each 
affording  an  independent  course  under  a  professor,  who  alone  is  respon- 
sible for  the  system  and  methods  pursued."  One  of  the  eight  original 
schools  was  that  of  mathematics,  pure  and  applied.  The  first  profes- 
sor of  mathematics  (from  1825  to  1827)  was  Thomas  Hewett  Key,  of 
England.  He  was  a  graduate  of  Trinity  College,  Cambridge.  Besides 
his  ability  as  a  mathematician,  he  possessed  great  classical  and  general 
attainments.  He  resigned  his  position  in  order  to  accept  the  professor- 
ship of  Latin  in  the  London  University. 

His  successor  was  Charles  Bonnycastle,  of  England,  who,  upon  Mr. 
Key's  resignation,  was  transferred  from  the  chair  of  natural  philosophy 
to  that  of  mathematics,  which  he  continued  to  fill  until  his  death,  in 
.1840.  He  was  the  son  of  John  Bonnycastle,  who  was  widely  known  in 
England  and  America  for  his  mathematical  text-books,  and  was  edu- 
cated at  the  Royal  Military  Academy  at  Woolwich,  where  his  father 
was  professor.  His  father's  books  exhibit  those  faults  which  were  com- 
mon to  English  works  on  mathematics  in  his  day.    It  is  fair  to  presume, 

*Dr.  Gessner  Harrison,  in  Duyckinck's  Cyclopedia  of  American  Literature;  Article, 
"  University  of  Virginia." 


192  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

however,  that  Charles  belonged  to  that  coterie  of  English  mathemati- 
cians of  which  Herschel,  Peacock,  Whewell,  and  others  were  members. 
and  which  introduced  the  Leibnitzian  notation  and  also  the  ratio  defini- 
tion of  the  trigonometric  functions  into  Cambridge.  At  the  University 
of  Virginia  he  enjoyed  the  reputation  of  a  man  of  great  ability  in  math- 
ematics and  of  broad  general  knowledge.  His  lighter  writings  indicate 
that  he  could  have  shone  also  in  the  fields  of  literature.  We  are  happy 
in  being  able  to  quote  the  following,  from  Dr.  James  L.  Cabell,  profes- 
sor of  physiology  and  surgery  at  the  University  of  Virginia :  * 

"  Though  apparently  an  earnest  and  enthusiastic  student  of  the  higher 
mathematics,  it  was  the  constant  habit  of  Professor  Bonnycastle  to  make 
extensive  and  varied  excursions  into  other  fields  of  study,  such  as  his- 
tory, metaphysical  philosophy,  and  general  literature.  I  remember  to 
have  seen  in  his  private  library  after  his  death  several  volumes  of 
works  on  moral  philosophy  with  copious  marginal  notes  written  by  him. 
I  recall  in  this  connection  the  fact  that  he  used  to  speak  with  emphasis 
and  some  indignation  on  the  absurd  charge  that  the  study  of  mathe- 
matics tends  to  render  its  votaries  insensible  to  the  force  of  probable 
evidence,  and  that  when  strict  mathematical  investigation  cannot  be 
had,  persons  whose  mental  discipline  has  been  secured  by  such  training 
become  either  obstinately  skeptical  or  wildly  credulous.  He  insisted 
that  all  onesided  training  had  a  natural  tendency  to  narrow  the  intel- 
lect and  that  this  applied  to  all  other  branches  of  learning  and  all  pro- 
fessional pursuits  as  well  as  to  mathematics.  The  obvious  remedy  lies 
in  a  liberal  and  broad  culture.  It  was  doubtless  with  a  view  to  enforce 
his  precepts  by  occasional  examples  that  he  was  in  the  habit  of  deliver- 
ing at  the  opening  of  each  session  of  the  university  a  popular  lecture, 
the  topics  of  which,  having  apparently  a  very  remote  connection  with 
mathematical  studies,  were  actually  suggested  by  some  recent  publica- 
tions in  the  department  of  general  literature.  These  addresses  were 
greatl}^  admired  by  the  crowds  of  young  men  who  attended  them,  includ- 
ing, in  addition  to  his  own  class,  representatives  from  all  the  other  de- 
partments of  the  university.  He  was  also  a  contributor  to  a  literary 
magazine  published  by  the  faculty  in  1828-29.  Some  of  his  articles 
were  stories  of  more  than  ordinary  merit  in  this  class  of  literary  pro- 
ductions, and  would  probably  have  made  his  fortune  if  such  magazines 
as  Harper's,  Scribner's,  etc.,  had  existed  at  that  day  with  a  competent 
development  of  public  taste. 

"The  only  distinct  impression  which  I  can  now  recall  as  to  Professor 
Bonnycastle's  method  of  teaching  has  reference  to  his  attempts  to  in- 
doctrinate his  pupils  at  every  stage  of  their  studies  with  the  philosophy 
and  essential  principles  of  the  subject  under  consideration.  At  that 
time  most,  if  not  all,  the  usual  text-books  and  all  the  school  teachers 
gave  only  rules  which  the  student  was  to  apply.  So  far  as  the  students 
knew,  these  rules  might  be  wholly  arbitrary.    Professor  Bonnycastle 

*  Letter  to  the  writer,  January  4,  1889. 


INFLUX  OF  FEENCH  MATHEMATICS.  •         193 

in'sisted  on  the  necessity  of  placing  the  student  in  a  position  to  recog- 
nize the  true  significance  of  every  principle  laid  down.  This  was  done 
by  oral  lectures  characterized  by  remarkable  lucidity  of  statement  and 
by  a  marvellous  fertility  of  striking  illustrations.  These  lectures  were 
fully  appreciated  by  the  better  sort  of  students  in  the  advanced  classes, 
but  were  thought  by  most  of  us  to  be  thrown  away  upon  the  younger 
and  less  ambitious  members  of  the  lower  classes.  The  general  verdict 
of  all  classes  of  hearers  ascribed  to  Mr.  Bonuycastle  genius  and  attain- 
ments of  the  highest  and  most  varied  character."* 

The  text-books  used  by  Bonnycastle  in  pure  mathematics,  in  connec- 
tion with  his  lectures,  were  the  Arithmetic,  Algebra,  and  Differential 
Calculus  of  Lacroix,  the  first  two  in  Farrar's  translation.  The  theory 
of  the  integral  calculus  was  taken  from  Young,  the  examples  from  Pea- 
cock's Collection.  In  geometry  he  used  his  own  work  on  Inductive 
Geometry  (1834). 

In  pure  mathematics  there  were  in  his  time  three  classes :  the  "  First 
Junior,"  "  Second  Junior,"  and  "  Senior."  "  Of  these  the  First  Junior 
begins  with  arithmetic;  but  as  the  student  is  required  to  have  some 
knowledge  of  this  subject  when  he  enters  the  university,  the  lectures 
of  the  professor  are  limited  to  the  theory,  showing  the  method  of  nam- 
ing numbers,  the  different  scales  of  notation,  and  the  derivation  of  the 
rules  of  arithmetic  from  the  primary  notion  of  addition ;  the  addition, 
namely,  of  sensible  objects  one  by  one.  The  ideas  thus  acquired  are 
appealed  to  at  every  subsequent  step,  and  much  pains  are  taken  to 
exhibit  the  gradual  development  from  elementary  truths  of  the  ex- 
tensive science  of  mathematical  analysis."  (Catalogue  for  1836.)  After 
a  thorough  course  in  arithmetic  students  were  well  prepared  for  alge- 
bra. In  teaching  the  rules  for  adding  and  subtracting,  etc.,  they  were 
compared  with  the  corresponding  rules  in  arithmetic,  and  the  agree- 
ment and  diversity  were  noticed  and  explained.  The  elements  of  geom- 
etry were  taught  and  illustrated  by  models.  The  book  on  Inductive 
Geometry  was  prepared  especially  for  the  use  of  his  students.  It  in- 
cludes geometry,  trigonometry,  and  analytical  geometry.  In  the  defi- 
nition of  the  trigonometric  functions  the  ratio  system  is  used.  "  The 
chief  result  which  the  author  hoped  to  secure  by  the  proposed  innova- 
tion was  such  an  arrangement  of  the  subject  as  would  enable  him  to 

*  In  another  part  of  Ms  letter  Dr.  Cabell  says:  "I  felt  bound  to  tell  you  that 
owing  to  my  complete  want  of  mathematical  knowledge,  even  to  the  extent  of  igno- 
rance of  the  terminology  of  the  science,  I  was  utterly  incompetent  to  form  a  critical 
judgment  of  Professor  Bonnycastle's  method  of  teaching.  I  can,  however,  recall  with 
some  vividness  the  impression  made  upon  me  at  the  time  when  he  caused  me  and  my 
fellow-students  to  understand  the  significance  of  processes  which  we  had  previously- 
applied  in  a  purely  arbitrary  method.  It  is  probable  that  we  exaggerated  the  merit 
of  our  new  professor  by  contrasting  him  with  the  very  imperfect  and  defective  stand- 
ards of  the  common  schools  of  Virginia  at  that  day.  I  believe,  however,  that  these 
defects  were  common  to  the  whole  country  when  Professor  Bounycastle  introduced  a 
reform  which  in  a  few  years  may  have  become  general." 
881— No,  3-^ 13 


194  TEACHING  AND   HISTORY    OF   MATHEMATICS. 

dispense  with  the  distinctions  hitherto  made  between  the  different 
branches  of  geometry,  and  thus  permit  him  to  treat  the  problems  em- 
braced under  the  heads  of  synthetic  geometry,  analytic  geometry,  and 
the  two  trigonometries,  as  composing  one  uniform  doctrine,  the  science 
of  Quantity  and  Position."* 

The  general  plan  appears  to  be  a  good  one,  in  the  main.  -  But  its  exe- 
cution is  not  satisfactory.  The  work  covers  631  crowded  pages.  The 
form  in  which  the  subject-  is  presented  is  bad.  Theorems  and  their 
demonstrations  are  in  the  same  kind  of  type,  and  the  eye  finds  nothing 
to  assist  and  relieve  it  in  passing  over  the  crowded  pages  of  prolix  ex- 
planations.   Nor  is  the  reasoning  always  good.t 

His  Inductive  Geometry  is,  we  believe,  the  only  mathematical  work 
which  he  published  while  he  was  professor  at  the  University  of  Vir- 
ginia. 

Both  algebra  and  geometry  were  begun  in  the  "  First  Junior"  class 
(catalogue  1836),  and  then  continued  in  the  "Second  Junior"  class.  Cal- 
culus was  begun  in  this  class  and  then  completed  in  the  "  Senior  "class. 

The  notation  of  Leibnitz  was  used  at  the  University  of  Virginia  from 
the  very  beginning. 

In  the  Virginia  Literary  Museum,  a  weekly  journal  issued  in  1829 
by  the  professors  of  the  university,  we  read  of  an  examination  of  the 

*  Preface  to  Inductive  Geometry. 

t  "Angles  are  so  evidently  portions  of  space  sarfounding  their  vertex,  and  this  space 
so  manifestly  the  same  in  all  cases,  that  we  are  forced  to  regard  it,  directly  or  indi- 
rectly, as  the  standard  to  which  all  angles  should  be  referred  "  (p.  112).  The  reason- 
ing by  which  the  sum  of  the  three  angles  of  a  triangle  is  shown  to  be  two  right 


angle3,isasfollow8(p.  123):  "Thelines^^,  C-D,  C£^,  *  *  *  that  enclose  a  small 
triangle  at  C,  are  separated  by  the  openings  o,  6,  c,  that  are  nearly  equal  to  the  angles 
of  the  triangle ;  two  of  these  openings,  namely,  a  and  c,  are  identical  with  angles  of 
the  triangle,  and  the  third,  t,  which  forms  a  space  indefinitely  extended,  differs  from 
the  opening  we  call  the  angle  C  merely  by  the  small  space  included  in  the  triangle. 
"  This  last,  by  bringing  the  triangle  nearer  to  C,  may  be  rendered  as  small  as  we 
please ;  and  thus  a  triangle  can  always  be  assigned  whose  angles  shall  differ  from  a, 
b,  c,  and,  consequently,  the  sum  of  whose  angles  shall  differ  from  two  right  angles 
by  less  than  any  assignable  quantity.  Some  difference  between  the  results  appears, 
it  is  true,  always  to  remain  ;  but  if  we  examine  more  attentively  the  idea  that  we 
are  able  to  form  of  infinite  space,  we  shall  find  the  difference  in  question  merely  ap- 
parent, and  shall  perceive  the  sum  of  the  three  angles  to  be  rigidly  equal  to  two  right 
angles."  This  reasoning  is  bad.  It  involves,  unnecessarily,  the  consideration  of  in- 
finite spaces. 


INFLUX  OF  FEENCH  MATHEMATICS.  196 

Seuior  class  in  mathematics,  on  Thursday,  July  16,  1829:  "The  mem- 
bers of  the  class  were  examined  in  application  of  algebra  to  geometry 
and  the  theory  of  curves,  as  contained  in  the  IV  chapter  of  Lacroix's 
Traite  dii  Calcul  Differentiel  et  du  Calcid  Integral.  In  the  differen- 
tial and  integral  calculus  they  were  examined  by  examples  taken  from 
the  questions  on  these  subjects  published  by  Peacock  &  Herschel. 
The  class  have  studied  the  differential  calculus  chiefly  from  the  treat- 
ise of  Boucharlat,  and  the  integral  from  Boucharlat,  Lacroix,  and  the 
examples  before  mentioned.  They  have  proceeded  to  the  integration 
of  partial  differential  equations  of  three  or  more  variables,  and  the  ques- 
tions proposed  were  chosen  to  this  extent."  These  extracts  show  that 
the  course  of  mathematics  taught  by  Professor  Bonnycastle  was  remark- 
ably far  advanced,  compared  with  the  work  done  in  the  ordinary  college 
or  university  in  this  country  at  that  time. 

Besides  the  three  classes  above  given  there  was  from  the  beginning 
a  class  in  mixed  mathematics  (really  a  graduate  class).  Under  Bonny- 
castle the  text- books  in  this  study  were  Venturalfs  Mechanics  and  the 
first  book  of  Laplace's  Mecanique  Celeste.  The  principles  were  applied 
to  various  problems.  A  separate  diploma  has  been  given  to  students 
completing  this  course  of  mixed  mathematics. 

Professor  Bonnycastle  left  a  large  number  of  mathematical  MSS.  in 
the  keeping  of  Professor  Henry,  of  the  Smithsonian  Institution,  who  a 
short  time  before  his  death  sent  them  to  be  deposited  in  the  library  of 
the  University  of  Virginia. 

After  the  death  of  Bonnycastle,  Pike  Powers,  now  a  minister  at  Eich- 
mond,  held  the  chair  until  J.J.  Sylvester  was  elected  professor,  inl841. 
Mr.  Powers  was  a  young  mathematician  of  fine  gifts  and  attainments, 
and  a  pupil  of  Bonnycastle.  Professor  Sylvester  was  then  already  gener- 
ally recognized  as  a  man  of  brilliant  genius  and  profound  mathemati- 
cal learning.  He  resigned  in  about  half  a  year,  and  afterward  ac- 
cepted a  professorship  in  the  Eoyal  Military  Academy  at  Woolwich. 
We  shall  have  to  say  more  about  him  in  connection  with  the  Johns 
Hopkins  University.  Prof  Pike  Powers  was  again  appointed,  tempo- 
rarily, to  teach  the  mathematics. 

The  next  possessor  of  the  mathematical  chair  was  Edward  H.  Court- 
enay,  from  1842  to  1853.  He  was  the  first  regular  occupant  of  this  chair 
who  was  educated  in  this  country.  He  was  born  in  Baltimore,  in 
1803.  After  having  been  examined  for  admission  to  the  U.  S.  Military 
Academy  at  West  Point,  in  1818,  the  examiner  remarked :  "  A  boy  from 
Baltimore,  of  spare  frame,  light  complexion,  and  light  hair,  would  cer- 
tainly take  the  first  place  in  his  class."  Courtenay  completed  the  four 
years'  course  in  three  years,  and  graduated  at  the  head  of  his  class  in 
1821.  From  that  time  till  1834  he  was  connected  as  teacher  with  the 
Military  Academy,  excepting  the  period  from  1824  to  1828.  After  leav- 
ing West  Point  he  was  for  two  years  professor  of  mathematics  at  the 
University  of  Pennsylvania,  then  he  became  division  engineer  for  the 


196  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

New  York  and  Erie  Eailroad.  He  was  employed  by  the  United  States 
Government  as  civil  engineer  ia  the  construction  of  Fort  Independence, 
Boston  Harbor,  from  1837  to  1841.  Just  before  his  appointment  to  the 
professorship  at  the  University  of  Virginia  he  was  chief  engineer  of 
dry-dock,  iNavy-Yard,  Brooklyn,  N.  Y.* 

Mr.  Courtenay  was  a  mathematician  of  noble  gifts  and  a  great  teacher. 
"  His  mind  was  quick,  clear,  accurate,  and  discriminating  in  its  appre- 
hensions, rapid  and  certain  in  its  reasoning  "processes,  and  far-reaching 
and  profound  in  its  general  views.  It  was  admirably  adapted  both  to 
acquire  and  use  knowiedge,"t  He  was  modest  and  unassuming  in  his 
manner,  even  to  dif&dence.  He  would  never  utter  a  harsh  word  to 
pupils  or  disparage  their  efforts.  "  His  pleasant  smile  and  kind  voice, 
when  he  would  say,  '  Is  that  answer  perfectly  correct  f '  gave  hope  to 
many  minds  struggling  with  the  difficulties  of  science,  and  have  left 
the  impression  of  affectionate  recollection  on  many  hearts.''^ 

Eegarding  his  work  at  the  University  of  Virginia,  Professor  Venable 
(at  one  time  a  pupil  of  Courtenay)  says  that  his  course  in  pure  mathe- 
matics was  prepared  and  written  out  (or  rather  printed  on  white  cloth 
in  large  letters)  with  great  care — following  Bonnycastle  in  the  use  of 
Young  in  the  treatment  of  the  differential  and  integral  calculus.  His 
course  in  this  branch  embraced  differential  equations  and  the  calculus 
of  variations.  His  MSS.  on  these  two  subjects  for  the  Senior  class  fill 
nearly  one  hundred  and  fifty  pages  of  his  printed  work.  His  notes  on 
the  calculus  were  published  in  1857,  after  his  death,  and  became  a 
valued  text-book  in  many  institutions.  ''In  its  publication  the  plan, 
language,  and  even  the  punctuation  have  been  followed  with  a  fidelity 
due  to  the  memory  of  a  friend."  The  work  was  more  extensive  than 
any  which  had  yet  appeared  in  this  country  on  the  same  subject.  Courte- 
nay added  descriptive  geometry  to  the  regular  course  of  pure  mathe- 
matics. He  i)repared  extensive  notes  for  his  class  in  mixed  mathematics, 
which  embraced  a  full  course  in  the  applications  of  the  calculus  to 
mechanics  and  to  the  planetary  and  lunar  theories  (perturbations). 

In  1845  the  course  in  the  School  of  Mathematics  was  as  follows : 
Junior  class,  theory  of  arithmetic,  algebra,  synthetic  geometry;  Inter- 
mediate class,  plane  and  spherical  trigonometry,  land  surveying,  navi- 
gation, descriptive  geometry  and  its  application  to  spherical  projection, 
shadows,  perspective ;  Senior  class,  analytical  geometry,  calculus.  The 
class  in  mixed  mathematics  studied  selections  from  Poisson,  Francceur, 
Pontecoulant,  and  others.  This  embraced  the  mathematical  investiga- 
tions of  general  laws  of  equilibrium  and  motion,  both  of  solids  and  fluids. 
The  text-books  for  that  year  were,  Lacroix's  Arithmetic,  Davies'  Bour- 
don, Legendre's  Geometry,  Davies'  Surveying  and  Descriptive  Geome- 
try, Davies'  Analytical  Geometry,  Young's  Differential  and  Integral 
Calculus. 

*Courtenay's  Calculus,  p.  iv.  t  Ibid.,  p.  v.  t  Hid.,  p.  vii. 


INFLUX  OF  FRENCH  MATHEMATICS.  197 

After  the  death  of  Courtenay  the  chair  of  mathematics  was  filled  by 
Albert  Taylor  Bledsoe.  He  was  a  native  of  Kentucky,  and  graduated 
at  West  Point  in  1830.  He  was  one  year  adjunct  professor  of  mathe- 
matics and  French  at  Kenyon  College,  Ohio ;  then  one  year  professor 
of  mathematics  at  Miami  University,  Ohio.  Afterward  he  practiced 
law  for  eight  years  at  Springfield,  111.  Before  his  coming  to  the  Uni- 
versity of  Virginia  he  was  professor  of  mathematics  and  astronomy  at 
the  University  of  Mississippi.  He  remained  in  his  new  position  till 
1863,  then  became  assistant  secretary  of  war  in  the  Southern  Confed- 
eracy. After  the  War  he  became  principal  of  a  female  academy  in  Bal- 
timore and  editor  of  the  Southern  Review.  He  died  in  1877  at  Alex- 
andria, Ya. 

Prof.  Francis  H.  Smith,  of  the  University  of  Virginia,  who  was  asso- 
ciated with  Bledsoe  in  the  faculty  of  the  institution,  writes  us  about 
him  as  follows :  "  He  succeeded  here  an  eminent  teacher,  Prof.  Edward 
H.  Courtenay  5  and,  while  the  two  men  were  most  unlike  in  every  respect, 
Dr.  Bledsoe's  evident  ability  so  impressed  his  class,  that  the  prestige 
of  the  mathematical  class  suffered  no  loss  in  his  hands.    From  his  life- 
long addiction  to  metaphysical  studies,  he  entered  with  great  zeal  upon 
the  philosophy  of  mathematics,  a  subject  which  every  infantile  mathe- 
matician is  bound  to  have  an  attack  of,  but  which  in  its  widest  rela- 
tions may  very  well  tax  the  powers  of  the  most  mature  and  advanced 
geometer.    In  this  field  I  think  Dr.  Bledsoe  won  a  place  by  the  side  of 
Bishop  Berkeley  and  Auguste  Comte.    His  treatise  on  the  Philosophy  of 
Mathematics  was  put  in  print  and  had  a  considerable  circulation.    He 
established,  a  new  course  of  lectures  here  in  connection  with  the  usual 
mathematical  curriculum,  upon  the  History  and  Philosophy  of  Mathe- 
matics.   That  feature  survives  to  this  day.    As  a  manipulator  of  mathe- 
matical formulae  and  solver  of  mathematical  problems,  Dr.  Bledsoe  was 
not  strikingly  able.    I  have  known  many  men  of  far  less  strength  who 
were  his  superiors  in  mere  algebraic  dexterity.    Yet,  I  was  convinced 
from  several  incidents  which  came  to  my  knowledge  during  his  teach- 
ing here  that  had  his  life,  after  he  left  West  Point,  been  devoted  to  the 
science,  he  would  have  left  the  pure  mathematics  simplified  in  statement 
and  improved  in  form.    His  originality  and  force  were  obvious  to  me, 
to  whom  he  freely  communicated  his  difficulties  and  successes,  during 
his  entire  residence  here.    I  learned  that  while  at  the  Military  Academy 
these  traits  were  strikingly  exhibited  by  his  solving  a  problem  in  the 
tangencies  of  circles  which  had  up  to  that  time  baffled  the  geometrical 
skill  of  the  academy,  and  which  had  been  left  unsolved  by  Archimedes' 
himself.    The  solution  given  by  Dr.  Bledsoe  was  afterward  published 
in  the  Southern  Eeview,  of  which  the  doctor  was  editor  and  pro  prietor 
for  a  number  of  years  before  his  death.    He  had  in  the  latter  y  earr<  of  his 
life  completed  a  treatise  on  synthetical  geometry,  of  the  Euclidian  type, 
and,  I  think,  had  found  a  publisher,  but  whether  it  ever  got  prin  ted  I 
am  not  aware.    Dr.  Bledsoe's  greatest  work  was  in  the  field  of  vLxetQi- 
physical  theology,  constitutional  laWj  and  review  articles. " 


198  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

His  Pliilosophy  of  Mathematics,  published  in  1867,  exhibits  brilliant 
controversial  powers.  lb  initiated  a  reactionary  movement  among  us 
against  the  unphilosophical  exposition  of  the  calculus  in  the  colleges  of 
our  land.  The  book  is  somewhat  verbose  in  its  style.  The  bulk  of  it 
consists  of  criticisms  of  various  text-books.  Comparatively  little  space 
is  given  to  what  the  author  considers  to  be  the  true  explanation  of  the 
subject.  It  seems  to  us  that  the  criticisms  which  he  makes  are  generally 
good  and  well  founded,  but  that  he  fails  in  proposing  a  sound  substitute 
for  the  explanations  which  he  rejects.  The  influence  of  the  book  has 
been  beneficial  in  so  far  as  it  has  caused  many  teachers  to  meditate 
upon  the  philosophy  of  the  calculus. 

He  gave  lectures  also  on  the  history  of  mathematics — a  subject  which 
received  little  or  no  attention  in  our  colleges  at  that  time.  He  prepared, 
but  never  published,  a  work  on  analytical  geometry,  in  which,  by  the 
discussion  of  one  equation  which  contained,  wrapped  up  within  itself, 
the  whole  folio  of  Apollonius  on  conic  sections,  he  developed  the  prop- 
erties of  the  circle,  ellipse,  hyperbola,  and  parabola.* 

Bledsoe  pursued,  in  the  main,  the  course  in  i)ure  mathematics  laid 
down  by  his  predecessor,  except  that  Courtenay's  Calculus  was  used  in 
place  of  Young's.  For  the  class  in  mixed  mathematics  he  used  (in  1854) 
Bartlett's  Analytical  Mechanics,  Newton's  Principia,  and  Pratt's  Me- 
chanical Philosophy.  Pontecoulant's  Systeme  du  Monde  was  also  used 
by  him  for  his  class. 

Professor  Bledsoe  was  not  very  strict  with  students  in  their  daily 
work,  but  on  approach  of  examination  day  he  knew  how  to  prepare  a 
tough  set  of  questions. 

By  temporary  appointment,  Alexander  L.  Nelson  taught  mathematics 
during  part  of  the  session  1853-54 ;  Robert  T.  Massie  during  part  of 
the  session  1861-62,  and  Francis  H.  Smith,  of  the  School  of  Natural 
Philosophy,  from  1863  to  1865. 

During  the  War  the  university  barely  subsisted ;  but  scarcely  was 
peace  restored  ere  the  institution,  amidst  perplexing  pecuniary  embar- 
rassments, prepared  with  resolute  energy  to  enlarge  its  capacity  for 
useful  work  by  multiplying  its  schools.  In  1867  the  School  of  Applied 
Mathematics  with  reference  to  Engineering  was  established. 

In  1865  Charles  S.  Venable  was  appointed  to  the  chair  of  mathe- 
matics, a  position  which  he  still  occupies.  He  is  a  native  of  Virginia, 
and  was  l:)orn  in  1827.  After  graduating  at  Hampden-Sidney  College 
in  1842,  he  remained  one  year  at  the  college  as  a  resident  graduate,  pur- 
suing mathematics  under  Col.  B.  L.  Ewell  (a  West  Point  graduate,  and 
afterward  president  of  William  and  Mary  College),  and  English  liter- 
ature -and  history  under  Maxwell.  He  then  became  tutor  in  mathe- 
matics, in  which  capacity  he  continued  two  years,  devoting  part  of  his 
time  to  the  study  of  law.  In  1845,  he  went  to  the  University  of  Vir- 
ginia and  spent  one  session  in  the  study  of  law,  mathematics,  and  ian- 

*  Bledsoe'e  Philosopli,y  of  Mathematics,  p.  130. 


INFLUX  OP  FRENCH  MATHEMATICS.  199 

guages.  Here  he  took  the  mathematical  lectures  of  Professor  Oourte- 
nay.  He  was  then  elected  professor  of  mathematics  at  Hampden-Sid- 
ney  College,  to  succeed  Ewell.  After  remaining  there  one  year  he 
obtained  leave  of  absence,  returned  to  the  University  of  Virginia,  and 
studied  mixed  mathematics  and  engineering  under  Professor  Courtenay. 
He  returned  to  Hampden- Sidney  in  1848,  and  filled  the  chair  of  mathe- 
matics till  June,  1852.  He  then  obtained  leave  of  absence  again,  and 
visited  Germany  for  the  further  prosecution  of  studies.  In  Berlin  he 
studied  astronomy  under  Encke,  and  mathematics  with  Dirichlet  and 
Borchardt.  He  then  went  to  Bonn,  studying  some  months  under  Pro- 
fessor Argelander,  the  director  of  the  observatory  of  Bonn.  While  iu 
Germany  astronomy  was  his  chief  branch  of  study.  He  then  travelled 
in  Southern  Europe,  studied  for  some  time  in  Paris,  visited  England, 
and  then  returned  to  Hampden-Sidney  College,  in  1853.  In  1856  he 
was  elected  to  the  chair  of  natural  philosophy  and  chemistry  at  the 
University  of  Georgia,  to  succeed  John  Le  Conte,  and  in  1857,  profes- 
sor of  mathematics  and  astronomy  in  the  South  Carolina  College.  In 
1858  he  published  an  edition  of  Bourdon's  Arithmetic.  Yenable  took 
part  in  the  attack  upon  Fort  Sumter,  and  took  active  part  in  the  War 
until  its  close.*  Since  his  connection  with  the  University  of  Virginia, 
Professor  Venable  has  issued  a  series  of  text-books,  consisting  of  First 
Lessons  in  Numbers,  1866,  revised  in  1870  j  Mental  Arithmetic,  1866 ; 
Practical  Arithmetic,  1867,  revised  in  1871 ;  Intermediate  Arithmetic, 
1872;  Elements  of  Algebra,  1869;  Elements  of  Geometry,  1875;  Notes 
on  (analytical)  Solid  Geometry. 

These  rank  among  the  best  and  most  rigorously  scientific  school- 
books  published  in  this  country.  In  his  arithmetics,  the  attempt  is 
made  "  to  render  the  reasoning  of  such  arithmetics  as  those  of  Bourdon, 
Briot,  DeMorgan,  and  Wrigley,  easily  accessible  to  the  young."  His 
Elements  of  Geometry  is  "after  Legendre,"  but  it  differs  from  the  orig- 
inal in  the  discussion  of  parallels,  in  the  use  of  the  methods  of  limits 
instead  of  the  method  of  the  reductio  ad  absurdum,  in  the  fuller  treat- 
ment of  certain  parts  of  the  subject,  and  in  giving,  at  the  beginning,  a 
chapter  on  the  Theory  of  Proportion  (in  which  the  theory  of  limits  is 
used  for  iucommensurables)  instead  of  presupposing  a  knowledge  of 
proportion,  as  is  done  by  Legendre.  One  feature  is  carried  out  in  this 
geometry  more  extensively  than  in  any  other  of  our  books,  namely,  the 
insertion  of  "hints  to  solutions  of  exercises."  A  teacher  who  does  not 
make  his  pupils  solve  original  problems  in  geometry,  is  a  failure.  But 
the  exercises  given  in  jnost  books  are  not  sufficiently  graded,  and  the 
young  beginner  is  very  apt  to  get  discouraged.  The  "  hints"  given  in 
this  book  serve  the  excellent  purpose  of  assisting  and  encouraging  the 
pupil  in  his  first  attempts  at  original  work.  In  1887  Professor  Venable 
published  an  Introduction  to  Modern  Geometry,  which  serves  as  an 

*Our  sketch  of  the  early  career  of  Professor  Venable  is  taken  from  La  Borde's  His- 
tory of  South  Carolina  College,  1874,  p.  474. 


200  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

appendix  to  liis  geometry.  The  treatment  of  the  subject  is  metrical 
rather  than  descriptive. 

The  method  of  instruction  under  Professor  Venable  has  be^n  essen- 
tially the  same  as  that  followed  by  his  predecessors.  It  consists  of  lect- 
ures, prelections  on  approved  text-books,  and  exercises  for  testing  and 
developing  the  power  of  the  student  in  original  solutions.  Great  stress 
is  constantly  laid  on  the  solution  by  the  student  of  original  exercises. 
In  this  respect,  each  meeting  of  the  cl  ass  is  a  seminarium.  In  delivering 
their  lectures,  some  professors  of  the  university  write  condensed  notes 
on  the  blackboard,  others  give  syllabuses.  The  students  very  soon  get 
up  printed  or  lithographed  notes  on  the  lectures.  The  practice  of 
reading  the  lectures  does  not  prevail  at  the  university. 

One  might  suppose  that  in  an  institution  where  students  have  the 
privilege  of  attending  whatever  school  they  please,  the  enrollment  in 
the  school  of  mathematics  would  be  comparatively  small.  This  has, 
however,  not  been  the  case  here.  The  attendance  on  this  school  is,  as  a 
rule,  greater  than  on  any  other  school  of  the  academic  department.  In 
three  or  four  sessions,  since  the  War,  the  number  of  students  in  the 
school  of  Latin  has  been  greater,  but  by  not  more  than  half  a  dozen 
students.  The  full  attendance  is  in  itself  good  evidence  of  the  careful 
teaching  and  efficient  work  in  the  mathematical  department.  In  order 
to  present  a  fuller  picture  of  the  services  of  Professor  Yenable,  we  quote 
from  a  letter  of  R.  H.  Jesse,  professor  of  Latin  at  the  Tulane  University 
of  Louisiana,  and  a  former  student  of  the  University  of  Virginia.  "In 
my  day  Colonel  Venable  was  absolutely  the  most  popular  among  the 
students  of  all  the  professors  in  the  University  of  Virginia.  At  the 
same  time  his  control  was  perfect  over  all  his  classes,  and  indeed  oyer 
any  and  all  bodies  of  students  with  whom  he  came  in  contact.  Doubt- 
less his  experience  as  an  officer  of  rank  in  the  Confederate  service,  his 
long  practice  in  teaching,  and  his  never  failing  kindness  of  heart  and 
sympathy  with  young  men,  produced  both  the  popularity  and  the  power 
of  control. 

"  Ever  since  I  have  known  the  institution  well,  now  nearly  twenty 
years,  he  has  been,  more  than  any  other  man,  active  and  able  in  pro- 
moting her  best  interests.  To  him  in  large  degree  was  due  the  increase 
by  the  State,  in  1875  or  1876  ,  of  her  annual  contribution  from  fifteen 
thousand  dollars  to  thirty  thousand  dollars.  This  increase  was  accom- 
panied with  the  condition  that  all  Virginia  students  able  to  pass  the 
entrance  examinations  to  the  academical  schools  should  be  educated  in 
those  schools  free  of  charge.  To  him  chiefly  was  due  the  raising  of  the 
endowment  fund  whereby  the  McCormick  telescope  was  gained  for  the 
university.  To  him  chiefly  has  been  due  the  large  increase  in  attend- 
ance upon  the  university  in  late  years.  Twice  he  has  been  Chairman 
[of  the  faculty]  and  twice  has  he  laid  the  office  down  voluntarily,  when 
the  university,  guided  safely  by  his  wisdom  and  energy  through  some 
serious  difficulties,  had  reached  excellent  condition  again.    He  has  had, 


INFLUX  OF  FRENCH  MATHEMATICS.  201 

to  my  certain  knowledge,  many  flattering  calls  to  other  fields,  far  more 
profitable  in  money,  but  lie  has  immediately  declined  them  all  to  stand 
fast  by  his  alma  maferJ'^ 

The  high  and  rigid  standard  inaugurated  by  Bonnycastle  and  Oourte- 
nay  has  been  rigorously  adhered  to.  The  standard  of  graduation  has 
always  been  high,  in  fact,  very  high  in  comparison  with  the  standards 
in  most  other  American  colleges.  The  mathematical  course  has  been 
broadened,  as  the  preparation  of  students  under  the  influence  of  the 
university  upon  the  academies  and  colleges  has  become  broader  and 
better.  "  We  have  many  excellent  preparatory  schools  in  Virginia," 
says  Professor  Venable,  "  which  prepare  students  well,  far  into  the 
differential  and  integral  calculus  in  such  works  as  Todhunter's  and 
Courtenay's  Calculus." 

The  course  in  mathematics,  as  stated  in  the  catalogue  for  1887-88,  is 
as  follows : 

I.  Puke  Mathematics. 

Jdnior  Class.— This  class  meets  three  times  a  week  (4|  hours)  and  studies  theory 
of  arithmetical  notations  and  operations ;  algebra,  through  the  binomial  theorem ; 
geometry,  plane  and  solid ;  geometrical  analysis,  with  numerous  exercises  for  original 
solution ;  elementary  plane  trigonometry,  embracing  the  solution  of  triangles,  with 
the  use  of  logarithms,  and  some  applications  to  problems  of  "heights  and  distances." 
The  preparation  desirable  for  it  is  a  good  knowledge  of  arithmetic,  of  algebraic  opera- 
tions through  equations  of  the  second  degree,  and  of  the  iirst  three  books  of  plane 
geometry. 

Textbooks— -ToihuntnT's  Algebra;  Venable's  Legendre's  Geometry,  witli  collection  of  exercises; 
Todhunter's  Trigonometry  for  Beginners. 

Intermediate  Class. — This  class  meets  twice  a  week  (3  hours)  and  studies  geo- 
metrical analysis,  with  exercises  for  original  solution;  plane  trigonometry,  with 
applications ;  analytical  geometry  of  two  dimensions ;  spherical  trigonometry,  with 
applications ;  elements  of  the  theory  of  equations.  The  preparation  desirable  for 
this  class  is  a  thorough  knowledge  of  algebra  through  the  binomial  theorem,  and 
logarithms ;  of  synthetic  geometry,  plane  and  solid,  with  some  training  in  the  solu- 
tion of  geometrical  problems ;  and  a  knowledge  of  the  elements  of  plane  trigonometry, 
including  the  use  of  logarithmic  tables. 

Tea;i-6oofcs.— Snowball's  Trigonometry,  Packle's  Conic  Sections,  the  Professor's  Collection  of  Exer- 
cises in  Plane  Geometry. 

Senior  Class. — This  class  meets  three  times  a  week  (4|  hours)  and  studies  analyt- 
ical geometry  of  three  dimensions,  through  the  discussion  of  the  conicoids  and  some 
curves  in  space ;  differential  and  integral  calculus,  with  various  applications ;  a  short 
course  in  the  calculus  of  variations ;  the  theory  of  equations,  and  lectures  on  the 
history  of  mathematics. 

Text-hooks. — The  Professor's  Notes  on  Solid  Geometry  (Analytical) ;  Todhunter's  Differential  Calcu- 
lus; Williamaon's  Integral  Calculus*;  .Todhunter's  Theory  of  Equations. 

Candidates  for  graduation  in  pure  mathematics  are  required  to  pursue  in  the  uni- 
versity the  studies  of  both  the  Intermediate  and  Senior  Classes. 

II.  Mixed  Mathematics. 

This  course  is  designed  for  those  students  who  may  desire  to  prosecute  theii* 
studies  beyond  the  limits  of  pure  mathematics.     It  embraces  an  extended  course  of 

*In  former  years  Professor  Venable  used  Courtenay's  Integral  Calculus,  which  was 
supplemented  with  notes  which  "  nearly  equalled  the  text."    (Prof.  R.  H.  Jesse.) 


202  TEACHINa  AND   HISTORY   OF   MATHEMATICS. 

reading  under  the  instruction  and  guidance  of  the  professor  on  the  applications  of  the 
diiferential  and  integral  calculus  to  mechanics,  physical  astronomy,  and  selected  por- 
tions of  physics.    The  class  in  mixed  mathematics  meets  twice  a  week  (3  hours). 
Text-looks Price's  Infinitesimal  Calcolas,  Vols.  II  and  HI ;  Cheyne's  Planetary  Theory. 

Mathematical  physics  and  spherical  astronomy  are  taught  in  the 
school  of  natural  philosophy,  in  charge  of  Prof.  F.  H.  Smith.  !N"orton's 
Astronomy  is  one  of  the  text-books.  In  this  school,  under  practical 
physics,  are  studied  also  the  method  of  least  squares. 

In  addition  to  theunder-graduate  course  in  mathematics  there  is  now 
a  more  extended  course,  occupying  a  large  part  of  two  sessions  of  nine 
months.  It  is  given  to  graduates  who  are  candidates  for  the  degree  of 
doctor  of  philosophy  in  the  mathematical  sciences.  This  course  in- 
cludes, in  addition  to  the  course  in  mixed  mathematics,  the  study  of 
modern  higher  algebra,  modern  higher  geometry  (Steiner's  or  some  like 
work),  a  fuller  study  of  the  differential  and  integral  calculus  (Price  and 
Hoiiel),  determinants  (taught  at  the  university  for  the  last  fifteen  years), 
a  fuller  course  in  differential  equations,  probabilities,  and  other  selec-' 
tions.  If  the  candidate  chooses  astronomy  for  his  secondary  branch,  then 
he  studies  Gauss's  Theoria  Ilotus,  and  enters  into  the  practical  compu- 
tation  of  orbits.  Should  he  choose  physics,  then  he  studies  some  of  the 
advanced  treatises  in  the  line  of  mathematical  physics. 

In  order  to  give  a  better  idea  of  the  course  leading  to  the  degree  of 
doctor  of  philosophy,  we  quote  from  a  letter  of  Dr.  S.  M.  Barton: 

^'This  doctorate  course  consisted  of  graduate  studies  in  pure  and 
mixed  mathematics  and  mathematical  and  practical  astronomy,  and  the 
text-books  read,  and  on  which  I  was  examined,  were  as  follows :  Hoiiel's 
Calcul  Infinitesimal,  four  volumes  ;  Chasles's  Traite  de  Geometrie  Supe- 
rieure ;  Price's  Infinitesimal  Calculus,  Vol.  Ill  (Statics  and  Dynamics  of 
Material  Particles)  5  Cheyue's  Planetary  Theoryj  Aldis's  Eigid  Dynam- 
ics ;  Notes  and  Examples  selected  by  the  Professor. 

"  The  above  were  required  in  the  mathematical  department.  In  as- 
tronomy the  text-books  and  requirements  were :  Gauss's  Theoria  Motus; 
Notes  on  the  Computation  of  Orbits,  by  Prof.  Ormond  Stone ;  Notes  on 
Least  Squares,  Pertuibations,Yariations  of  Constants,  etc.,  by  Professor 
Stone;  Computation  of  the  Orbit  of  Barbara  (No.  234).  This  last  was 
of  course  a  work  of  several  months. 

"  I  was  allowed  to  select  my  own  subject  for  a  thesis,  which  was  ac- 
cepted by  the  faculty  and  printed  before  1  stood  my  last  examinations. 

"In  the  preparation  for  this  thesis  I  was  obliged  to  read,  outside  of 
the  studies  laid  down  in  the  course,  the  method  of  equipollences,  and 
the  principles  of  quaternions,  and  various  articles  bearing  on  the  sub- 
ject, in  which  I  made  use  of  the  following  works:  Exposition  de  la 
Metliode  des  Equipollences,  by  Bellavitis,  translated  into  French  by 
Laisaiit.  La  Vraie  Theorie  des  Quantites  Negatives,  etc.,  by  Mourey. 
Articles  in  the  Nouvelles  Annales  de  Mathematiques.  Kelland  a,nd  Tait's 
Introduction  to  Quaternions.    Tait's  Quaternions. 


INFLUX  OF  FRENCH  MATHEMATICS.  20S 

"  In  pursuing  these  doctorate  studies  I,  of  course,  made  use  of  many- 
books  for  reference,  among  which  I  might  mention  Salmon's  Conic  Sec- 
tions and  Higher  Plane  Curves,  and  Geometry  of  Three  Dimensions. 
Gregory's  Examples.  Vols.  I  and  II  of  Price's  Calculus.  Some  older 
works  by  Peacock  and  others,  as  well  as  some  more  elementary  trea- 
tises.   *    *    * 

"  I  can  not  refrain  *  *  *  from  alluding  to  one  striking  feature 
of  the  mathematical  teaching  at  the  University  of  Virginia,  namely, 
independence  in  the  student;  and  by  independence  I  mean  the  spirit  of 
self-reliance  which  enables  the  student  to  work  out  and  elucidate  for 
himself. 

"  The  student  is  taught  from  the  start  to  depend  upon  himself. 

"  This  spirit  of  self-reliance  pervades  the  mathematical  department, 
and  it  promotes  originality,  as  well  as  gives  zest  to  the  work. 

"  This  would  seem  to  be  the  only  true  way  to  teach  mathematics,  but 
many  of  our  elementary  teachers  do  little  or  nothing  to  inculcate  this 
great  ijrinciple." 

The  thesis  referred  to  above  is  entitled  "  Bella vitis's  Method  of  Bqui- 
pollences"  (1885).  It  contains  an  outline  of  the  calculus  of  equipollences 
and  of  its  relation  to  quaternions.  It  shows  that  while  equipollences 
are  more  readily  mastered,  and  yield  on  the  whole  more  expeditious 
solutions  of  plane  problems  than  quaternions,  the  latter  are  immeasur- 
ably superior  in  elegance,  logical  simplicity,  and  extent  of  application. 

Since  Professor  Yenable  has  been  connected  with  the  University  of 
Yirginia,  the  department  of  mathematics  has  graduated  many  students 
who  have  become  prominent  as  teachers  and  scientists  in  their  specialty. 
Chief  among  these  are  Prof.  C.  E.  Vawter,  professor  of  mathematics  in 
Emory  and  Henry  College  for  some  years,  now  in  charge  of  the  Miller 
Manual  Training  School ;  Prof.  G.  Lanza,  professor  of  mathematics  at 
the  Massachusetts  Institute  of  Technology ;  Prof.  W.  M.  Thornton,  of 
the  school  of  applied  mathematics,  University  of  Virginia ;  Professor 
Graves,  professor  of  mathematics  at  the  University  of  E'orth  Carolina ; 
Professor  Gore,  professor  of  physics  and  astronomy  at  the  University 
of  ISTorth  Carolina ;  Professor  Bohanuan,  professor  of  mathematics  at 
the  University  of  Ohio  (Columbus);  Prof.  H.  A.  Strode,  principal  of 
Keumore  University  High  School,  Virginia;  Prof.  W.  H.  Echols,  pro- 
fessor of  engineering  and  president  of  the  school  of  mines  at  the  Uni- 
versity of  Missouri ;  Prof.  W.  H.  Richancer,  professor  of  mathematics 
at  the  school  of  mines,  University  of  Missouri ;  Prof.  T.  U.  Taylor,  as- 
sistant professor  of  mathematics,  University  of  Texas. 

Applied  mathematics,  i.  e.,  mathematics  applied  to  civil  engineering, 
was  taught  in  the  school  of  mathematics  almost  at  the  beginning  of  the 
university.  In  1832  a  class  in  engineering  was  organized  as  a  separate 
department  under  the  professor  of  mathematics,  and  was  maintained  as 
an  attachment  to  the  school  of  mathematics  until  1850.  It  was  then 
left  out  of  the  catalogue  from  the  fact,  no  doubt,  that  the  successful 


204  TEACHING   AND    HISTOEY    OF   MATHEMATICS. 

working  of  such  a  course  imposed  too  heavy  a  burden  upon  the  mathe- 
matical professor.  In  1865  the  department  of  civil  engineering  was 
revived  and  placed  under  the  joint  charge  of  the  professors  of  mathe- 
matics, physics  and  chemistry.  In  1867  Prof.  Leopold  J.  Boeck  was 
made  assistant  professor  and  placed  in  charge  of  the  school  of  applied 
mathematics,  comprising  courses  in  civil  and  mining  engineering. 
These  led  to  the  degrees  of  civil  and  mining  engineer,  respectively. 
In  1868  Professor  Boeck  was  promoted  to  the  full  professorship  of 
applied  mathematics.  He  held  the  chair  until  1875,  when  he  resigned, 
and  was  succeeded  by  Wm.  M.  Thornton,  as  assistant  professor.  Pro- 
fessor Thornton  was  subsequently  promoted  to  the  full  professorship 
of  applied  mathematics.  This  school  has  sent  out  a  large  number  of 
engineers  of  sound  training. 

Mention  should  be  made  here  of  the  school  of  practical  astronomy, 
under  the  direction  of  Prof.  Ormond  Stone.  He  is  also  director  of  the 
McGormick  Observatory,  and  editor  of  the  Annals  of  Mathematics. 

UNIVERSITY  OP  NORTH  CAROLINA.* 

Professor  Mitchell's  successor  in  the  chair  of  mathematics  was  James 
Phillips,  from  1826  to  1867.  Professor  Love  speaks  of  him  as  follows : 
"  He  was  born  in  England  in  1792.  It  is  not  known  at  what  school  he 
received  his  early  education.  The  greater  portion  of  his  mathematical 
education  was  gotten  by  private  study.  He  came  to  America  in  1818 
and  opened  an  academy  in  Harlem,  E".  Y.  Here  he  won  reputation  as 
an  instructor,  and  by  contributions  to  the  mathematical  publications  of 
the  day.  In  1826  he  came  to  North  Carolina  as  professor  of  mathe- 
matics and  natural  philosophy. 

"  He  was  a  patient  student  of  the  masters  in  mathematics,  of  Fergu- 
son, Newton,  Delambre,  Laplace,  and  others.  He  prepared  a  text- 
book on  conic  sections  which  was  published  and  used  as  an  introduction 
to  analytic  geometry.  He  left  in  manuscript  the  greater  portion  of  a 
series  of  text-books  on  mathematics,  including  the  calculus.  These 
were  most  carefully  prepared,  but  for  some  reason  he  never  published 
any  of  them.  Probably  the  War  was  the  cause  of  his  not  publishing. 
He  left  directions  when  he  died  that  all  his  MSS.  should  be  burned. 
Among  them  were  also  many  translations  from  French  mathematical 
works. 

"  That  Dr.  Phillips  never  published  more  is  very  much  to  be  regretted. 
He  had  great  mathematical  ability,  and  was  an  extremely  careful  and 
lucid  writer.  Like  Dr.  Mitchell,  he  divided  his  time  and  energy.  Both 
of  them  were  ministers  and  spent  much  time  in  the  preparation  of  ser- 
mons. Dr.  Phillips  left  hundreds  of  manuscript  sermons;  and  these  he 
directed  to  be  burned  with  all  his  other  MSS.    He  died  suddenly  of 

*  For  all  the  information  here  given  ou  the  University  of  North  Carolina,  the  writer 
is  indebted  to  Prof.  James  L.  Love,  associate  professor  of  mathematics  at  the  uni- 
versity. 


INFLUX  OF  FEENCH  MATHEMATICS.  205 

apoplexy  iu  the  college  chapel,  where  he  had  gone  to  conduct  morning 
prayers,  on  the  14th  of  March,  1867." 

The  requirements  for  admission  were  raised  in  1835  so  as  to  include 
all  of  arithmetic.  It  seems  that  in  the  same  year  a  little  of  algebra — 
"Young's  Algebra  to  simple  equations"— was  also  required.  The  in- 
crease in  the  requisites  for  entering  college  were  brought  on  at  this  time 
with  excessive  haste,  and  we  are  not  surprised  that,  after  three  years' 
trial,  algebra  was  withdrawn.  It  was  not  again  required  until  1855, 
when  candidates  were  examined  on  "  algebra  through  equations  of  the 
first  degree."    No  alterations  were  made  till  1868. 

As  regards  the  courses  of  study,  Professor  Love  says :  "  In  1835 
arithmetic  was  dropped,  algebra  was  coinpleted  in  the  Freshman  year, 
and  conic  sections  and  analytic  geometry  begun  in  the  Sophomore  year. 
In  1839  mechanics  was  introduced  into  the  Sophomore  and  Junior  years, 
civil  engineering  into  the  Senior  year,  and  since  that  date  analytic 
geometry  has  been  completed  in  the  Sophomore  year.  Calculus  was 
begun  in  the  Sophomore  year  in  1841,  and  from  that  date  to  1868  it  was 
sometimes  in  the  Sophomore  year  and  sometimes  in  the  Junior  year. 
For  fifty  years,  from  1818  to  1868,  first  fluxions  and  then  differential  and 
integral  calculus  were  required  of  all  graduates.  A  three-years'  course 
in  engineering  was  introduced  in  1854.  It  included  in  addition  to  the 
regular  course  required  for  graduation,  descriptive  geometry,  drawing, 
shades  and  shadows,  mechanics,  civil  engineering,  and  geodesy.  This 
course  was  continued  until  1862. 

"An  attempt  was  made  in  1855  to  offer  some  election  of  courses  in  the 
Sophomore  and  Junior  years.  Two  courses  were  offered,  the  one  analyt- 
ical, the  other  geometrical.  The  latter  embraced  geometry,  plane  and 
spherical  trigonometry,  mensuration,  surveying,  navigation,  natural 
philosophy,  and  astronomy.  The  analytical  course  included,  in  addi- 
tion, analytical  geometry,  differential  and  integral  calculus,  statics  and 
dynamics,  acoustics  and  optics.  During  the  Freshman  year  the  two 
courses  were  identical,  but  for  the  Sophomore  and  Junior  years  different 
text-books  were  used,  even  for  the  same  subjects,  in  the  two  courses. 
After  two  years'  trial,  these  double  courses  were  given  up.  From  1857 
to  1868  the  one  mathematical  course  was  as  follows :  Freshman  year, 
algebra,  geometry;  Sophomore  year,  plane  and  spherical  trigonometry 
with  applications,  analytical  geometry,  differential  and  integral  calculus; 
Junior  year,  natural  philosophy  and  astronomy," 

Our  list  of  books  used  by  Professor  Phillips  is  quite  complete ;  Ryan's 
Algebra  was  used  in  1827 ;  Young's  Algebra  was  introduced  in  1836; 
Peirce's  was  studied  from  1844  to  1868.  In  geometry,  Legendre  was 
used  for  a  time.  About  1843  Peirce's  Geometry  was  introduced,  and 
not  dropped  till  1868,  except  for  the  years  1855  to  1857,  when  Perkins 
and  Loomis  were  used  each  one  year.  From  1857  to  1868,  Munroe's 
"  Geometry  and  Science  of  Form"  was  used  in  the  Freshman  class  as  an 
introduction  to  geometry.    The  idea  of  premising  a  course  in  demon- 


206  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

strati ve  geometry  by  a  short  one  in  empirical  geometry  is  very  com- 
mendable. In  descriptive  geometry,  Davies'  was  introduced  in  1854; 
also  his  Shades  and  Shadows.  In  1844  Peirce's  trigonometry  was  in- 
troduced; Perkins's  was  used  from  1855  to  1856;  Charles  Phillips's 
from  1856  to  1860 ;  Loomis's  from  1860  to  1868.  In  conic  sections  James 
Phillips's  was  taught  from  1830  to  1847,  when  Peirce's  book  was  intro- 
duced. From  1851  to  1868  Loomis's  was  studied,  except  from  1853  to 
1855,  when  Church's  and  Smith's  Biot's  were  used,  each  a  year. 

In  calculus  the  notation  of  Leibnitz  was  introduced  in  1830.  Hut- 
ton's  work  was  supplanted  in  1847  by  Peirce's  Curves,  Functions,  and 
Forces,  which  was  followed  in  1851  by  Loomis's.  This  was  used  until 
1868,  except  in  1853,  when  Church's  was  taught  for  one  year.  In  as- 
tronomy, Peirce's  book  was  introduced  in  1847,  Herschel's  in  1855,  and 
Norton's  in  1857.  ^ 

Before  the  Civil  War  the  university  was  prosperous  and  popular. 
The  courses  in  mathematics  described  above  were  certainly  very  credit- 
able for  their  day.  Hon.  Wm.  H.  Battle  spoke  of  the  university  as  fol- 
lows :  "  In  the  extent  and  variety  of  its  studies,  the  number  and.  ability 
of  its  instructors,  and  the  number  of  its  students,  it  surpassed  nearly 
all  similar  institutions  in  our  own  section  of  the  country,  and  was  be- 
ginning to  rival  the  old,  time-honored  establishments  of  Yale  and 
Harvard.  In  the  year  1858  its  catalogue  showed  a  larger  number  of 
under-graduates  than  that  of  any  other  college  in  the  United  States, 
except  Yale.  All  this  success  was  accomplished  in  a  very  short  time. 
A  glance  at  the  rapidly  increasing  ratio  of  its  graduates  will  illustrate 
the  truth  of  my  remark.  For  the  first  ten  years  after  the  date  in  which 
degrees  were  conferred  by  the  university,  the  number  of  students  who 
received  the  baccalaureate  was  53  ;  for  the  second  decade  it  was  110; 
for  the  third,  259 ;  for  the  fourth,  146 ;  for  the  fifth,  308 ;  for  the  sixth, 
448 ;  and  for  the  seventh  the  annual  number  was  going  on  at  a  rate 
which  would  have  produced  882,  nearly  the  double  of  that  which  im- 
mediately preceded  it."  * 

During  the  Civil  War  nearly  all  Southern  colleges  closed  their 
doors,  but  not  so  the  University  of  North  Carolina.  It  was  the  boast 
of  its  president  that  "  during  the  four  years  of  war  the  college  bell  never 
failed  in  its  daily  calls,  that  the  faculty  was  ever  in  place  for  duty,  and 
'that  all  grew /at  on  sorghum  and  corn  bread;'  that  the  institution 
was  maintained  in  full  icorking  order.^^  The  severest  blow  to  the  pros- 
perity of  the  university  came  after  the  War.  In  1868  the  old  faculty 
was  turned  out  by  the  "  reconstructed  "  State  government,  and  from 
1869  to  1871  a  new  faculty  labored  to  make  the  university  popular 
again.  But  political  feeling  was  too  high;  the  university  was  closed 
from  1871  to  1875. 

*Addres8  delivered  before  tlie  two  literary  societies  of  the  University  of  North 
Carolina,  June  1, 1865,  by  Hon.  Wm,  H.  Battle. 


INFLUX  OF  FKENCH  MATHEMATICS.  207 

Charles  Phillips  became  professor  of  mathematics  in  1875.  He  had 
been  tutor  from  1844  to  1853,  associate  professor  from  1855  to  1860,  and 
professor  of  engineering  from  1853  to  1860.  In  1879  he  was  made  pro- 
fessor emeritus  of  mathematics,  and  Ealph  H.  Graves,  jr.,  who  had 
been  professor  of  engineering  since  1875,  became  now  professor  of 
mathematics.  Professor  Graves  is  a  graduate  of  the  University  of  Vir- 
ginia and  a  former  pupil  of  Professor  Venable.  Since  1885  James  Lee 
Love  has  been  associate  professor  of  mathematics.  He  graduated  at 
the  university  at  the  head  of  his  class,  and  then  took  a  graduate  course 
in  mathematics  at  the  Johns  Hopkins  University  in  the  year  1884-85. 

Under  the  present  able  corps  of  instructors,  mathematical  teaching 
is  again  flourishing.  Since  the  reopening,  in  1875,  the  requirements  in 
mathematics  for  admission  have  been  :  arithmetic,  and  algebra  to  quad- 
ratic equations.  The  course  in  mathematics  has  been  as  follows  :  Fresh- 
men, algebra,  geometry  ;  SopJwiores,  plane  and  spherical  trigonometry, 
logarithms,  plane  analytical  geometry ;  Juniors,  theory  of  equations, 
differential  and  integral  calculus,  natural  philosophy ;  Seniors,  mechan- 
ics, astronomy.  The  studies  of  the  first  and  second  years  have  been 
required  of  all  graduates.  The  studies  of  the  third  year,  except  natural 
philosophy,  have  been  elective.  Mechanics  and  astionomy  were 
required  in  all  courses  leading  to  degrees  until  1885.  Since  that  time 
mechanics  is  elective  in  all  courses,  and  astronomy  elective  in  the  A.  B. 
course.  Since  1885  postgraduate  electives  have  been  offered  in  solid 
analytic  geometry  (Smith's),  determinants,  differential  equations,  mod- 
ern algebra,  and  quaternions.  From  1875  to  1879  a  three-year  course 
in  engineering  was  offered.  Since  1879  the  course  has  been  partially 
withdrawn  ;  and  at  present  (1888)  it  includes  only  a  one-year  course  in 
surveying,  descriptive  geometry,  and  projective  drawing. 

Eobinson's  University  Algebra  was  used  from  1869  to  1871,  and  since 
1875  Schuyler's,  Venable's,  Newcomb's,  and  Well's — Kewcomb's  most. 
In  geometry  the  books  have  been,  since  1875,  those  of  Venable,  Went- 
worth,  Newcomb,  and  J.  W.  Wilson.  In  descriptive  geometry  and  pro- 
jective drawing  Warren's  is  taught.  Davies'  Trigonometry  was  used 
from  1869  to  1871,  Wheeler's  since  1875,  and  Kewcomb's  since  1882. 
In  calculus  the  works  of  Peck,  Courtenay,  Bowser,  Byerly,  and  Tod- 
hunter  have  been  in  use.  Since  1883  Williamson  has  been  the  text- 
book. Newcomb  and  Deschanel  are  the  books  in  astronomy  and  phys- 
ics. 

In  1883  the  Elisha  Mitchell  Scientific  Society  was  organized.  The 
professors  of  mathematics  take  part  in  its  exercises.  Meetings  are  held 
once  each  mouth  for  the  presentation  of  papers  on  any  scientific  subject. 
The  society  publishes  a  Journal,  with  abstracts  of  the  more  important 
papers  read,  and  the  writer  has  before  him  Vol.  V,  Part  I,  in  which 
appear  two  papers  by  Professor  Graves  on  geometrical  subjects.  These 
have  been  published  also  in  the  Annals  of  Mathematics,  to  which  Pro- 
fessor Graves  is  a  frequent  contributor. 


208  TEACHING   AND   HISTOKY   OF   MATHEMATICS.     ' 

UNIVERSITY  OF  SOUTH  CAROLINA.* 

The  successor  of  Eev.  Dr.  Hanckel  in  the  chair  of  mathematics  was 
James  Wallace.  He  entered  upon  his  duties  in  1820,  and  remained  at 
the  college  for  fourteen  years.  Some  years  previous  to  his  coming  to 
this  institution  he  had  been  professor  at  Georgetown  College,  in  West 
Washington.  He  possessed  mathematical  ability  and  fine  attainments 
in  his  specialty.  While  at  Columbia,  S.  C,  he  contributed  to  the  South- 
ern Eeview  articles  on  "Geometry  and  Calculus,"  Vol.  1 5  "Steam 
Engine  and  Eailroad,"  Vol.  VII ;  "  Canal  Navigation,"  Vol.  VIII.  In 
the  first  of  the  above  articles  a  somewhat  severe  criticism  of  Hassler's 
Trigonometry  is  given.  Wallace  upholds  the  geometrical  method  and 
the  line  system.  He  contributed  also  to  Silliman's  Journal,  in  one  num- 
ber, giving  an  account  of  a  new  algebraic  series  of  Stainville  in  Ger- 
gonne's  Annals,  but,  by  mistake,  it  was  not  duly  accredited,  and  ap- 
peared like  Wallace's  work.  This  drew  him  into  a  controvorsy  with 
Nathaniel  Bowditch. 

Wallace's  ability  is  shown  by  his  treatise  on  the  Use  of  the  Globes 
and  Practical  Astronomy  (New  York,  1812).  This  work  was  in  advance 
of  any  other  American  treatise  on  astronomy  of  its  day.  The  work  had 
512  pages,  was  printed  closely,  with  lengthy  notes  in  small  type.  Some 
parts  required  little  or  no  knowledge  of  mathematics  on  the  part  of  the 
reader ;  others  assumed  a  knowledge  of  geometry,  trigonometry,  conic 
sections,  and  algebra,  and  the  last  part  also  of  fluxions.  The  title 
page  bears  the  motto,  "  Quid  munus  Beipubliccv  majus  aut  melius  afferre 
possinms,  quam  si  Inventutem  hene  erudiamus  ? — Cicero." 

M.  La  Borde  says  in  his  History  that  Wallace  did  not  place  very 
high  value  upon  the  above  work.  "  He  said  the  MS.  of  a  work  to 
which  he  had  devoted  twenty  years  of  his  life  was  destroyed  by  fire, 
and  he  thought  that  but  for  that  accident  he  would  have  left  something 
worthy  of  remembrance." 

As  a  teacher  Wallace  was  in  some  respects  the  opposite  of  Blackburn. 
The  latter  was  somewhat  hot-tempered,  but  Wallace  was  a  patient  and 
laborious  teacher,  who  loved  his  art.  "  No  obtuseness  of  perception, 
no  degree  of  stolidity  could  provoke  him  to  ill-temper."  Upon  leaving 
the  college  he  retired  to  a  small  farm  near  Columbia,  where  he  died  in 
1851. 

After  the  departure  of  Wallace,  Lewis  R.  Gibbs  held  a  temporary  ap- 
pointment for  one  year,  or  part  of  one.  In  1835  Thomas  S.  Twiss  was 
appointed.  He  occupied  the  chair  for  eleven  years.  He  was  born  in 
Troy,  N.  Y.,  graduated  at  West  Point,  and,  before  his  election,  was 
teaching  a  classical  school  at  Augusta,  Ga.  He  was  remarkable  for  in- 
dustry, punctuality,  and  "  watching  and  waiting  "  to  catch  students  in 
mischief.    He  enjoyed  the  reputation  of  arraigning  more  offenders  than 

*  The  material  for  this  sketch  was  kindly  furnished  us  by  Prof.  E.  W.  Davis,  pro- 
fessor of  mathematics  and  astronomy  at  the  university. 


INFLUX  OF  FRENCH  MATHEMATICS.  209 

any  other  two  members  of  the  faculty.  Upon  leaving  the  college  he  be- 
came president  of  some  iron  works  in  the  Spartanburg  district.  From 
here  he  returned  to  his  old  home  in  'Sew  York. 

The  next  mathematical  professor,  Matthew  J.  Williams,  was  likewise 
a  West  Point  graduate  (class  of  1821).  He  was  a  native  of  Georgia, 
and  had  an  early  bent  for  arithmetic.  At  the  Military  Academy  he  was 
one  of  four  to  attain  a  maxluium  mark  in  mathematics.  In  1825  he  was 
stationed  at  Old  Point  Comfort,  Ya.,  then  at  Fort  Howard,  Wis.  He 
resigned  from  the  Army  in  1828,  and  studied  law  in  St.  Louis.  He  prac- 
ticed law  in  Georgia  until  1835,  when  he  received  an  appointment  to  the 
South  Carolina  Conference  of  the  Methodist  Episcopal  Church,  at  Cokes- 
bury,  Abbeville  County.  Thence  he  was  called  to  the  South  Carolina 
College.  He  resigned  in  1853  on  account  of  severe  disease.  His  health 
failed  during  his  last  year  at  West  Point,  and  he  seems  to  have  had  a 
constant  struggle  with  sickness  from  that  time  on.  As  a  teacher,  he 
was  "  zealous,  industrious,  and  thorough."  His  enthusiasm  knew  no 
bounds.  He  was  esteemed  as  a  scholar,  a  man,  and  a  Christian.  When 
his  health  began  to  decline  and  there  was  fear  he  would  have  to  give  up 
his  work,  the  presi(Jent  of  the  college  wrote  in  his  report  to  the  trustees: 
"  I  can  not  express  to  you  how  much  I  value  his  services  in  the  depart- 
ment which  he  fills,  and  I  should  regard  it  as  a  most  deplorable  calamity 
to  the  college  to  be  deprived  of  his  labors." 

His  successor,  Charles  F.  McCay,  a  Pennsylvauian,  was,  at  the  time 
of  his  election,  a  professor  at  the  University  of  Georgia,  and  a  colleague 
of  John  and  Joseph  Le  Conte.  He  was  elected  president  of  the  South 
Carolina  College  in  1855.  In  an  attempt  to  act  as  a  "  go-between  "  in  a 
disagreement  between  faculty  and  students,  he  incurred  the  displeasure 
of  both  parties.  After  his  resignation  he  went  into  business,  and  is  now 
actuary  of  an  insurance  company  in  Baltimore.  From  what  we  can 
learn,  he  was  a  man  of  ability  and  a  good  teacher. 

From  1857  to  1862  Charles  S.  Yenable  filled  the  chair  of  mathematics. 
Since  the  War  he  has  been  for  nearly  a  quarter  of  a  century  professor 
at  the  University  of  Virginia,  and  has  established  for  himself  a  lasting 
reputation  as  a  teacher  of  mathematics.  While  professor  at  Columbia, 
he  was,  as  yet,  a  young  man,  and  was  not  so  popular  as  a  teacher. 

We  proceed  to  give  the  courses  of  study  for  the  period  preceding  the 
War.  In  1836  the  terms  for  admission  were,  "  arithmetic,  including 
fractions  and  the  extraction  of  roofs."  In  1848  was  added,  "  algebra  to 
equations  of  the  first  degree."  In  1851  Davies'  Bourdon  was  the  algebra 
used.  In  1853  the  whole  of  Bourdon's  Algebra  was  required  for  en- 
trance. This  requisition  appears  to  have  been  excessive,  and  in  1859  it 
was  reduced  to  "  Bourdon's  Algebra  to  Chapter  IX  "  (thus  omitting  the 
general  theory  of  equations  and  Sturm's  theorem),  or  ^^  Loomis's  Algebra 
to  Section  XVII"  (omitting  permutations,  combinations,  series,  loga- 
rithms, and  general  theory  of  equations).  The  catalogues,  from  1857  to 
1862,  contain  this :  "A  thorough  knowledge  of  arithmetic  being  essential 
881— No.  3 14 


210  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

to  success  in  all  classes  of  the  college,  applicants  must  be  prepared  for 
a  full  and  searching  examination  in  this  study." 

In  1836  the  course  of  study  was  as  follows : 

^^Freshman  year :  Bourdon's  Algebra  to  equations  of  the  third  degree, 
ratios  and  proportions,  summation  of  infinite  series,  nature  and  con- 
struction of  logarithms,  Legendre's  plane  geometry.  Sop]io7nore  year : 
Legendre's  solid  geometry,  constructions  of  determinate  geometrical 
equations,  Davies'  mensuration  and  surveying,  including  methods  of 
plotting  and  calculating  surveys,  measurement  of  heights  and  distances, 
and  use  of  instruments  in  surveying.  Junior  year  :  Descriptive  geom- 
etry and  conic  sections,  principles  of  i^erspective,  analytic  geometry-, 
fluxions — direct  and  inverse  methods  in  their  application  to  maxima, 
minima,  quadrature,  cubature,  etc.  Senior  year:  Natural  philosophy 
and  astronomy. 

"  There  shall  be  daily  recitations  of  each  class,  one  after  morning 
prayers,  one  at  11  A.  m.,  one  at  4  p.  m.  On  Saturday  morning  there 
shall  be  one  recitation." 

In  the  introduction  of  descriptive  geometry  into  the  course,  we  no- 
tice West  Point  influences.  The  "fluxions"  above  mentioned  must 
mean  "differential  and  integral  calculus."  Mr.  Twiss,  the  professor  at 
this  time,  was  a  graduate  of  the  Military  Academy  at  West  Point,  and 
was  not  likely  to  teach  fluxions  and  the  Newtonian  notation. 

In  1838  the  Freshmen  finished  the  whole,  both  of  algebra  and  geom- 
etry 5  the  Sophomores  had  plane  and  spherical  trigonometry  in  place  of 
solid  geometry. 

In  1841  Davies'  Calculus  was  studied  in  the  Junior  year.  Three  years 
later,  the  Sophomores  were  taught  from  Davies'  works  on  Mensuration 
and  Surveying, Analytical  Geometry,and  Descriptive  Geometry.  In  1848 
(M.J.  Williams, professor)  Loomis's  Conic  Sections  were  studied.  De- 
scriptive geometry  and  calculus  were  taught  by  lectures.  After  complet- 
ing the  calculus,  in  the  Junior  year,  Olmsted's  Mechanical  Philosophy 
was  taken  up.  The  Seniors  had  courses  in  astronomy  and  (Mahan's) 
civil  engineering.  Owing  to  a  rise  in  the  terms  for  admigsiou  Bourdon's 
algebra  was  omitted  in  the  first  year,  the  studies  for  the  other  classes 
remaining  the  same.  In  1854  descriptive  geometry  was  thrown  out 
of  the  course.  Professor  McCay  was  not  a  West  Point  graduate,  and 
attached,  probably,  less  importance  to  this  branch.  In  1857  spherical 
geometry  was  transferred  from  the  second  to  the  third  year. 

In  1858  the  Freshmen  studied  geometry  (Legendre),  reviewed  algebra 
(applications  of  algebra  to  geometry) ;  the  Sophomores,  mensuration, 
surveying  and  leveling,  conic  sections  (Loomis),  mechanics  (gravity, 
laws  of  motion) ;  the  Juniors  had  lectures  on  calculus,  spherical  trigo- 
nometry, mechanical  philosophy  (Olmsted) ;  the  Seniors,  astronomy, 
civil  engineering,  natural  philosophy  (Olmsted). 

In  1860  Professor  Venable  introduced  at  the  end  of  the  first  year 
theoretical  arithmetic,  using  his  own  edition  ot  Bourdon.    He  used  also 


INFLUX  OP  FRENCH  MATHEMATICS.  211 

Loomis's  Geometry  in  place  of  Legendre.  In  1861  Loomis's  Geometry  is 
mentioned,  "  with  original  problems."  Algebra  was  reviewed  and  ap- 
plied to  "geometrical  problems."  We  judge  that  extra  efforts  were 
made  by  Professor  Venable  to  improve  On  tlie  traditional  methods  of 
teaching,  by  requiring  the  student  to  do  a  great  deal  of  original  work 
in  the  line  of  solving  problems. 

In  18G3  the  buildings  of  the  college  "  were  taken  possession  of  by  the 
Confederate  Government,  and  used  as  a  hospital  until  the  close  of  the 
War."  Its  charter  was  amended  by  the  Legislature  in  1865,  and  in  the 
following  year  it  was  re-opened  as  the  University  of  South  Carolina. 

The  mathematical  chair  was  given  to  B.  P.  Alexander,  a  graduate  of 
West  Point  and  a  man  of  great  ability.  During  the  War  he  was  a  Con- 
federate brigadier,  distinguished  himself  at  Gettysburg,  and  introduced 
"  signalling  "  into  the  Confederate  army.  As  a  teacher  he  was  much 
liked.  He  was  very  practical  and  to  the  point  in  his  methods  and  illus- 
trations. Since  leaving  the  college,  he  has  been  connected  with  rail- 
roads, either  as  president  or  otherwise. 

Prof.  T.  E.  Hart,  a  graduate  of  Heidelberg,  taught  mathematics  from 
1870  to  1872.  He  was  then  and  is  now  in  very  poor  health,  suffering 
from  paralysis.  While  he  was  professor  his  classes  bad  often  to  go  to  his 
house  for  recitation. 

From  1873  to  1876  A.  W.  Curamings  held  the  mathematical  chair.  At 
this  time  the  college  passed  through  the  darkest  period  of  its  history. 
These  were  the  unfortunate  years  of  "  reconstruction."  In  addition  to 
the  numerous  obstacles  which  American  colleges  generally  have  had  to 
encounter,  the  colleges  in  the  South  have  had  to  contend  with  great  polit- 
ical upheavals.  Like  the  University  of  IS  orth  Carolina,  the  University 
of  South  Carolina  closed  its  doors.  From  1876  to  1880'  the  institution 
was  without  faculty  and  without  students. 

When  the  institution  opened,  in  1866,  its  course  of  study  was  remod- 
eled. ,  In  this  reorganization  the  plan  of  the  University  of  Virginia 
was  followed.  In  the  prospectus  we  read  that  "  the  university  consists 
of  eight  schools ;"  that  students  are  allowed  to  cho(fse  the  departments 
which  they  wish  to  pursue,  provided  they  enter  at  least  three  schools. 
In  certain  cases,  however,  students  will  be  allowed  to  enter  lesis  thaii 
three  schools." 

The  prospectus  continues,  as  follows :  "During  the  present  year  there 
will  be  no  examinations  or  other  requirements  foi^  admission,  except 
that  the  applicant  must  be  at  least  fifteen  years  of  age ;  but  in  order  to 
ensure  uniformity  of  preparation  in  certain  departments,  a  preparatory 
course  has  been  prescribed,  and  after  this  year  applicants  (under  eight- 
een years  of  age)  will  be  required  to  bring  a  satisfactory  certificate  of 
proficiency,  or  to  stand  an  examination.  For  applicants  over  eighteen 
years  of  age,  no  examination  or  certificate  will  be  required  during  the 
next  year," 


212  TEACHING  AND   HISTOEY   OF   MATHEMATICS. 

"  In  all  the  different  schools  the  method  of  instruction  is  by  means  of 
lectures  and  the  study  of  text-books,  accompanied  in  either  case  by  rigid 
daily  examinations." 

In  the  "  school  of  mathematics,  and  civil  and  military  engineering 
and  construction,"  the  requirements  for  admission  were  :  "Arithmetic 
in  all  its  branches,  including  the  extraction  of  square  and  cube  roots ;" 
"  algebra,  through  equations  of  the  second  degree." 

From  1867  to  1872  the  terms  were  as  above,  together  with  "a  knowl- 
edge of  the  first  four  books  of  geometry,"  which,  "though  not  indis- 
pensable, is  very  desirable.'^ 

In  1872  the  management  of  the  university  fell  under  the  Eeconstruc- 
tion  administration ;  negroes  were  admitted,  and  a  four  years'  prepara- 
tory course  was  given.    The  catalogue  of  1872-73  says  : 

"  In  arithmetic,  attention  should  be  paid  to  all  the  rules  and  calcula- 
*3  tions  usually  given  in  written  arithmetic,  and  too  much  importance  can 
not  be  paid  to  a  thorough  preliminary  drill  in  mental  arithmetic." 

In  the  "college  of  literature,  science,  and  the  arts,"  the  requirements 
are,  in  addition,  for  the  classical  course,  "algebra,  as  far  as  equations  of 
the  second  degree,"  and  it  is  "  recommended  that  they  also  master  the 
first  four  books  in  Davies'  Legendre,  or  the  equivalent ;"  for  the  scien- 
tific course,  "  algebra,  up  to  radical  quantities." 

In  the  catalogue  for  1876,  the  requirements  were  "  the  whole  of  arith- 
metic,''  and  "  algebra  as  far  as  equations  of  the  second  degree." 

The  course  of  study  in  mathematics  was,  in  1866,  algebra  from  equa- 
tions of  the  second  degree  to  general  theory  of  equations  and  loga- 
rithms, geometry,  plane  and  spherical  trigonometry,  surveying  and  the 
use  of  instruments,  in  the  first  year ;  in  the  second  year,  descriptive 
geometry,  analytical  geometry,  calculus,  mathematical  drawing.  Text- 
books :  Loomis's  books  on  algebra  and  geometry,  Davies'  Shades,  Shad- 
ows, and  Perspective,  Church's  Analytical  Geometry  and  Calculus. 
In  the  "  department  of  mechanical  philosophy  and  astronomy,"  Prof. 
John  Le  Conte's  Mechanics  was  taught,  also  Olmsted's  Astronomy, 
witb  Herschel's  O^lines  and  Norton's  Astronomy  for  reference.  In 
1867  Loomis's  Astronomy  was  used,  as  well  as  his  series  of  mathemati- 
cal text-books  from  his  Algebra  to  his  Calculus.  In  1870  everything 
is  the  same  as  given  above,  except  that  mechanical  philosophy  and 
astronomy  were  temporarily  taught  by  the  professor  of  mathematics. 

In  1872  Robinson's  University  Algebra  and  Loomis's  Geometry  were 
studied  in  the  first  year ;  Robinson's  Trigonometry,  Mensuration,  Sur- 
veying, and  Spherical  Trigonometry  in  the  second  year;  Robinson's 
Analytical  Geometry  and  Conic  Sections  the  third  year.  Later  on 
Ficklin's  Algebra  was  introduced. 

"  In  1879  the  trustees  of  the  university  were  empowered  by  act  of 
the  General  Assembly  to  establish  a  College  of  Agriculture  and  Me- 
chanics at  Columbia,  and  to  use  the  property  and  grounds  of  the  col* 
lege  for  this  purpose.    This  was  accordingly  done  in  1880," 


INFLUX  OF  FRENCH  MATHEMATICS.  213 

"  In  1881  the  Legislature  granted  an  annual  appropriation  for  the 
support  of  the  schools  of  the  university,  and  in  1882  the  South  Caro- 
lina College  was  reorganized  by  the  appointment  of  a  full  faculty.  It 
went  into  active  operation  the  fall  of  the  same  year." 

From  1882  to  1888  Benjamin  Sloan  was  the  professor  of  mathematics. 
At  present  he  is  professor  of  physics  and  civil  engineering.  He  is  a 
South  Carolinian,  graduated  at  West  Point  in  1860,  served  in  K"ew 
Mexico  before  the  War,  and  then  entered  into  the  Confederate  service. 
The  story  goes  that  when  he  entered  upon  the  duties  of  his  chair  at  the 
college,  he  ordered  a  bookseller  to  get  Courtenay's  Calculus.  "  Calcu- 
lus ! "  replied  the  bookseller,  "  what  are  you  going  to  do  with  it  ■? " 
"  Teach  it,"  was  the  reply.  "  You  can't  do  that,  no  South  Carolina  boy 
ever  studies  calculus."  Though  this  be  merely  the  opinion  of  a  jovial 
bookseller,  it  is,  we  fear,  not  without  some  truth  when  applied  to  the  ten 
years  preceding  the  reorganization  and  re-opening  of  the  college  in  1882. 
For  four  years  it  was  under  Eeconstruction  rule,  and  for  six  years  its 
doors  were  closed  to  students. 

Professor  Sloan  is  a  first-class  teacher.  He  requires  a  great  deal  of 
original  work  of  students,  and  inspires  considerable  enthusiasm.  .In 
his  manner  he  is  very  quiet  and  easy.  Among  the  students  he  is 
liked  and  popular. 

In  1888  Dr.  E.  W.  Davis  was  elected  to  the  mathematical  chair.  He 
graduated  at  the  University  of  Wisconsin  in  1879,  and  after  spending 
some  time  at  the  Washington  Astronomical  Observatory,  went  to  the 
Johns  Hopkins  University,  where  for  four  years  he  studied  mathe- 
matics under  Professor  Sylvester  and  his  associates.  As  a  subsidiary 
study  Davis  pursued  physics  under  Professor  Hastings.  At  this  great 
university  he  soon  caught  the  spirit  and  enthusiasm  which  is  so  con- 
tagious there.  His  mind  was  chiefly  bent  toward  geometrical  studies, 
and  the  papers  from  his  pen,  which  are  published  in  the  Johns  Hop- 
kins University  Circulars  and  the  American  Journal  of  Mathematics 
are  evidences  of  his  power  as  an  original  investigator.  Before  his  ap- 
pointment to  his  present  position  he  was  professor  of  mathematics  for 
four  years  at  the  Florida  Agricultural  College  in  Lake  City.  In  his 
teaching  Professor  Davis  possesses  great  power  in  causing  students  to 
think.  He  is  a  bold  advocate  of  greater  freedom  from  formalism  in 
mathematical  instruction. 

The  terms  for  admission  on  the  re-opening  of  the  institution  were,  in 
mathematics,  arithmetic,  and  algebra  through  equations  of  the  first 
degree.  Eadicals  were  added  in  1883.  In  1884  the  terms  were,  arith- 
metic, and  algebra  to  equations  of  th6  second  degree.  No  additions 
have  been  made  since. 

The  mathematical  course  in  1882  consisted,  in  the  first  year,  in  the 
study  of  Newcomb's  Algebra,  Chauvenet's  Geometry  (six  books)  j  in 
the  second  year,  in  the  study  of  Kewcomb's  Plane  and  Spherical  Trig- 
onometry, Puckle's  Conic  Sections ;  in  the  third  year,  in  t  he  further  study 


214  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

of  conic  sections  (Puckle,  Olney),  and  calculus  (Olney,  Todhunter).  In 
applied  mjitbematics  courses  were  given  in  the- second  year  on  survey- 
ing (Gillespie)  and  drawing,  Peck's  Mechanics,  Wood's  Strength  and 
Eesistance  of  Materials,  and  Walton's  Problems  in  Elementary  Me- 
chanics, astronomy  (Loorais,  Kewcomb,  and  Hoiden),  and  Mahan's  Civil 
Engineering.  In  1884  Warren  became  the  text-book  in  descriptive 
geometry.  In  3885  Taylor's  Calculus  was  introduced;  in  1886  Watson's 
Descriptive  Geometry  and  Merriman's  Least  Squares ;  ih  1887  2!few» 
comb's  Analytic  Geometry. 

The  mathematical  text-books  for  1888  are,  in  the  first  year,  Todhun- 
ter's  Algebra  for  Beginners,  Byerly's  Chauvenet's  Geometry;  in  the  sec- 
ond year,  Blaklie's  Plane  and  Spherical  Trigonometry,  Peirce's  Tables; 
in  the  third  year,  Taylor's  Calculus,  Church's  Descriptive  Geometry;  in 
the  fourth  year,  Newcomb  and  Holden's  Shorter  Course  in  Astronomy. 

This  year  (1888-89)  a  graduate  department  has  been  added.  In 
mathematics  it  offers  the  following  branches:  Algebra  (theory  of  equa- 
tions, theory  of  determinants,  etc.),  geometry  (projective  geometry, 
higher  plane  curves,  etc.),  calculus  (differential  equations  and  finite 
differences),  elliptic  function::-,  astronomy,  and  quaternions. 

TJNIVEESITT  OF  ALABAMA.* 

The  University  of  Alabama  was  opened  in  1831,  with  Gurdon  Salton- 
stall  in  charge  of  the  mathematical  teaching.  Two  years  later  William 
W.  Hudson  became  professor  of  mathematics,  and  held  the  position  until 
1837,  when  Frederick  Augustus  Porter  Barnard  became  connected  with 
the  institution,  and  had  charge  of  the  mathematical  department  till 
1849.  The  wonderful  activity  of  this  powerful  man  in  the  various  de- 
partments of  science  gave  a  great  stimulus  to  higher  education  in  the 
State.  He  had  previously  been  tutor  at  his  ahna  mater^  Yale.  In  1849 
he  assumed  the  duties  of  the  chair  of  chemistry  at  the  University  of 
Alabama.  While  connected  with  the  institution  as  professor  of  mathe- 
matics and  natural  philosophy  he  wrote  and  published  an  arithmetic, 
which  came  for  a  time  into  pretty  general  use  in  Alabama.  In  1846  he 
was  appointed  astronomer  by  the  State,  to  settle  a  boundary  dispute 
between  Alabama  and  Florida.  He  was  appointed  astronomer  for  the 
State  of  Florida  also,  so  that  be  represented  both  States  in  the  settle- 
ment of  the  dispute.  Professor  Barnard  was  always  fond  of  mathe- 
matics. He  has  written  a  number  of  valuable  articles  on  mathematical 
subjects  for  Johnson's  New  Universal  Cyclopaedia. 

By  old  students  Professor  Barnard  is  always  spoken  of  in  most  laud- 
able terms.  Says  Dr.  B.  Manly :  "  To  me  the  study  of  physics,  astronomy, 
etc.,  under  Prof.  F.  A.  P.  Barnard,  ♦  *  *  and  of  chemistry  and 
kindred  sciences  under  Prof.  R.  H.  Bramby,  long  deceased,  were  the 

•  Nearly  all  the  material  for  tliis  article  was  sent  us  by  Prof.  T.  W.  Palmer,  pro- 
fessor of  mathematics  at  the  university. 


INFLUX  OF  FKENCH  MATHEMATICS.  215 

most  attractive  parts  of  my  college  course."  Mr.  John  A.  Foster,  now 
chancellor  of  the  south-eastern  chancery  division  of  Alabama,  was  a 
student  and  then  a  tutor  of  mathematics  at  the  university  in  the  time 
•that  Barnard  taught  there.    He  says : 

*'  I  entered  the  Sophomore  class  of  the  IJniversity  of  Alabama  at 
Tuscaloosa  in  the  autumn  of  1844,  and  received  my  diploma  in  August, 
1847,  in  a  class  of  eighteen.  During  my  college  course  Prof.  F.  A.  P. 
Barnard  was  the  professor  of  mathematics  and  John  G.  Barr  was  the 
assistant  professor  of  mathematics.  Dr.  Barnard  afterward  became 
the  president  of  the  University  of  Mississippi,  and  in  1861,  being  a 
Union  man,  resigned  and  went  North,  where  he  was  for  some  time 
engaged  in  the  scientific  department  of  the  Government,  and  afterward 
.  was  president  of  Columbia  College  in  the  city  of  Kew  York.  A  very 
short  time  ago  I  observed  that  he  has  retired  from  this  work. 

"  Professor  Barnard  was  not  less  distinguished  as  a  scientist  than  as 
a  mathematician.  His  reputation  is  world  wide.  I  was  a  great  friend 
of  his,  and  up  to  1858  I  was  a  constant  correspondent  with  him.  I  need 
hardly  say  that  his  instruction  was  thorough  and  far  in  advance  of  the 
methods  which  prevailed  at  that  time.  There  has  never  been  a  better 
teacher  of  mathematics,  and  those  now  living  still  claim  that  the  country 
is  but  now  getting  to  the  methods  of  teaching  practiced  by  him  more 
than  forty  years  ago.  Withal,  he  was  a  warm  and  generous  friend,  and 
was  very  popular  with  those  who  were  his  pupils.  During  the  summer 
of  1844  or  1845  he  went  to  Europe  and  spent  some  time  in  France,  and 
on  his  return  to  the  university  he  brought  with  him  the  newly  discovered 
Daguerrean  process,  and  took  pictures  experimentally  before  his  class. 
He  was  hard  of  hearing  and  had  a  deep  guttural  voice,  but  no  one  had 
a  happier  faculty  of  making  himself  clearly  understood.  He  married 
an  English  lady  while  I  was  his  pupil. 

"  Oapt.  John  G.  Barr,  the  assistant  professor,  was  worthy  to  occupy 
the  position  as  second  to  this  distinguished  man.  In  1847  he  raised  a 
company  and  went  to  the  Mexican  War,  where  he  served  with  distinc- 
tion until  its  close.  Soon  after  he  was  appointed  to  a  diplomatic  posi- 
tion by  the  United  States  Government,  and  died  at  sea  when  on  his  way 
out  to  assume  the  duties  of  his  official  station.  He  was  an  able  and 
successful  teacher  of  mathematics." 

Mr.  Foster  engaged  in  educational  work  till  1859  (being  for  some 
years  president  of  a  college  in  La  Grange,  Ga.),  when  he  went  to  the 
practice  of  law. 

The  mathematical  teaching  at  the  university  for  the  three  years  suc- 
ceeding 1849  was  in  the  hands  of  Prof.  Landon  Cabell  Garland,  now 
the  honored  chancellor  and  professor  of  natural  philosophy  and  astron- 
omy of  Vanderbilt  University.  His  successors  as  instructors  of  mathe- 
matics at  the  University  of  Alabama,  before  the  War,  were  Profs. 
George  Benagh  (1852-60),  Eobert  Kennon  Hargrove  (1855-57),  James 
T.  Murfee  (1860-61),  and  William  Jones  Vaughn  (1863-65). 


216  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

Prof.  E.  K.  Hargrove,  after  teaching  mathematics  for  two  years, 
joined  the  ministry  of  the  M.  E.  Church  South,  and,  a  few  years  ago, 
was  elected  bishop  by  the  general  conference  which  met  at  Nashville, 
Tenu. 

The  terms  for  admission  to  the  university  were,  1833-56,  arithmetic; 
1857-59,  arithmetic,  and  algebra  through  equations  of  the  second  de- 
gree; 1860-62,  arithmetic,  and  algebra  to  equations  of  the  first  degree; 
the  records  for  the  next  three  years  are  lost. 

Down  to  1852  the  professor  of  mathematics  was  at  the  same  time  pro- 
fessor of  physics,  according  to  the  usual  custom  in  American  colleges 
at  that  day.  In  1833  the  Freshman  class  completed  algebra  (Colburn? 
Lacroix)  and  commenced  geometry  (Farrar's  Legendre) ;  the  Sophomore 
class  studied  geometry,  trigonometry,  and  conic  sections.  The  Junior 
and  Senior  classes  were  taught  mechanics,  statics,  heat,  light,  elec- 
tricity, etc.  The  books  used  were  the  Cambridge  Mathematics  of  Pro- 
fessor Farrar.  This  course  continued  without  change  until  1842,  when 
surveying,  mensuration,  etc.,  were  made  an  importantpart  of  the  Sopho- 
more work.  In  1843  Davies'  text-books  were  adopted.  In  1845  Peirce's 
Algebra  was  introduced,  but  after  two  years  it  was  displaced  by  Davies'. 
In  1849  the  calculus  was  added  to  the  Junior  course.  The  text  used 
was  Church's  until  1855,  when  Loomis's  was  adopted.  From  1860  to 
1865  the  records  are  so  incomplete  that  it  is  impossible  to  state  whether 
or  not  any  changes  were  made  during  that  time. 

Before  the  War,  the  university  was  prospering.  "In  the  Junior  and 
Senior  classes,"  says  Mr.  Foster  (class  of  1847),  "much  attention  was 
given  to  applied  mathematics.  Physics,  astronomy,  surveying,  and 
navigation  were  taught.  The  university  was  but  a  college  with  a  fine 
corps  of  professors,  and  presented  advantages  offered  by  very  few  other 
institutions  of  learning  at  that  time." 

The  War  naturally  interfered  with  the  successful  working  of  the  uni- 
versity. In  1865  the  university  buildings  were  destroyed  by  fire,  and 
the  institution  was  not  opened  again  until  1869.  The  condition  of  the 
country  at  that  time  was  not  favorable  for  the  advancement  of  educa- 
tion. In  recent  years,  however,  decided  and  encouraging  progress  has 
been  made.  A  thrill  of  aspiration  and  enthusiasm  has  been  running 
through  Southern  colleges. 

The  first  year  after  the  re-opening  Prof.  IS".  E.  Chambliss  taught  the 
mathematics;  the  next  year.  Prof.  J.  D.  F.  Eichards;  and  the  year  fol- 
lowing. Prof.  Hampton  S.  Whitfield,  and  the  fourth  year  Prof.  David 
L.  Peck.  In  1872-73  Prof.  W.  J.  Vaughn  held  the  mathematical  chair; 
and  from  1873  to  1878  Prof.  H.  S.  Whitfield  again.  In  1878  Professor 
Vaughn  assumed  the  duties  of  this  chair  for  the  third  time,  and  dis- 
charged them  for  four  years.  Since  1882  Prof.  Thomas  Waverly  Palmer 
has  filled  the  chair  and  taught  with  marked  success. 

Vaughn  is  now  professor  of  mathematics  at  Vanderbilt  University. 
"  Though  he  has  never  written  text-books,"  says  Professor  Palmer,  "still 


INFLUX  OF  FEENCH  MATHEMATICS.  217 

he  is  justly  regarded  as  one  of  the  ablest  mathematicians  in  our  Ameri- 
can colleges."  Prof.  J.  K.  Powers,  president  of  the  Alabama  State 
Normal  School,  who  studied  at  the  university  from  1871  to  1873,  says 
that  he  had  completed  the  course  in  pure  mathematics  before  going  there, 
and  that  he  took  applied  mathematics  there.  "Prof,  Wm.  J.  Vaughn 
at  that  time  filled  the  chair  of  applied  mathematics.  He  was  (and  is) 
an  accomplished  mathematician,  an  attractive  instructor,  a  fine  general 
scholar,  and  a  charming  gentleman.  At  that  time  the  chair  of  pure 
mathematics  was  filled  by  Prof.  D.  L.  Peck  and  Prof.  H.  S.  Whitfield. 
I  Tcnew  nothing  of  their  methods,  but  pure  mathematics  was  not  popular 
in  those  days.  In  after  years,  when  Prof.  Yaughn  assumed  control  of 
that  work,  no  department  of  the  university  was  more  popular." 

Of  Professor  Palmer,  Chester  Harding  (class  of  '84,  now  a  cadet  at  the 
U.  S.  Military  Academy  at  West  Point)  says :  "  This  gentleman,  a  grad- 
uate of  the  class  of  '81  of  the  university,  had  so  satisfactorily  filled  the 
position  of  assistant  professor  during  the  preceding  term,  that  his  elec- 
tion was  secured,  as  young  as  he  was,  against  the  claims  of  other  appli- 
cants of  extensive  experience,  reputation,  and  influence." 

From  1869  to  1871  only  the  elements  of  arithmetic  were  required  for 
admission.  During  the  next  two  years,  algebra  to  equations  of  the 
second  degree  was  added.  In  1873  the  requirements  were  reduced  to 
arithmetic  alone.  ISo  change  was  made  until  1878,  when  algebra  through 
equations  of  the  second  degree  was  required.  Gradual  changes  have 
been  made  every  year  since,  and  now  the  whole  of  algebra  and  three 
books  of  geometry  are  required. 

The  catalogue  for  1887-88  states  that  the  candidate  for  admission 
"  must  pass  a  satisfactory  examination  in  arithmetic,  in  algebra  through 
arithmetical  and  geometrical  progression,  and  in  the  first  two  books  of 
geometry.  The  examination  in  arithmetic  will  include  the  whole  sub- 
ject as  embraced  in  such  works  as  White's,  Robinson's,  GofPs,  Greenleaf  s, 
or  Sandford's  higher  arithmetic.  In  algebra,  particular  stress  will  be 
placed  upon  the  use  of  parentheses,  factoring,  highest  common  factor, 
lowest  common  multiple,  simple  and  complex  fractions,  simple  equa- 
tions with  one  or  more  unknown  quantities,  involution,  evolution,  theory 
of  exponents,  radicals  (including  rationalization,  imaginary  quantities, 
properties  of  quadratic  surds,  square  root  of  a  binomial  surd,  and  solu- 
tion of  equations  containing  radicals),  quadratic  equations,  equations 
of  the  quadratic  form,  simultaneous  quadratic  equations,  ratio  and  pro- 
portion, arithmetical  and  geometrical  progression." 

In  1874  the  calculus  was  dropped  from  the  university  course,  but  was 
introduced  again  in  1878. 

In  1881  there  was  a  reorganization  of  the  courses  of  study.  Two 
courses  of  mathematics  were  arranged,  one  for  classical  and  scientific 
students,  and  one  for  engineering  students.  The  course  for  classical  or 
scientific  students  embraced  algebra,  geometry,  plane  and  spherical 
trigonometry,  and  analytic  geometry.    These  subjects  were  completed 


218  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

in  the  Sophomore  class.  Since  1881  no  changes  have  been  made  in  the 
classical  and  scientific  courses. 

The  engineering  course  embraced  all  subjects  that  were  taught  in  the 
classical  and  soieatific,  but  to  the  Sophomore  work  was  added  descrip- 
tive geometry,  and  to  the  Junior  class  calculus.  This  course  has  been 
modified  since.  At  present  it  consists  of  higher  algebra  and  geometry 
for  the  Freshmen ;  plane  and  spherical  trigonometry,  analytical  geom- 
etry, descriptive  geometry,  theory  of  equations,  for  the  Sophomores ; 
calculus,  determinants,  and  quaternions  for  the  Juniors. 

Determinants  and  quaternions,  which  are  regularly  in  the  course  since 
1887,  have  been  taught  irregularly  for  several  years.  Quaternions  are, 
according  to  catalogue,  now  taught  in  the  third  term  of  the  Sophomore 
year,  before  the  completion  of  analytic  geometry.  This  is  a  somewhat 
new  departure  in  the  arrangement  of  mathematical  studies,  and  one 
which  is  worthy  of  respectable  and  thoughtful  consideration. 

As  to  text-books,  in  1871  Davies'  Algebra  and  Geometry  were  used; 
also  Church's  Analytic  Geometry  and  Calculus.  In  1872  and  1873  the 
books  were  Robinson's  Algebra  and  Geometry,  and  Loomis's  Trigo- 
nometry and  Analytic  Geometry.  In  1878  Peck's  Analytic  Geometry 
and  Calculus  were  introduced.  The  books  used  at  present  are  Well's 
Algebra,  Wentworth's  Geometry,  Trigonometry,  and  Analytic  Geom- 
etry, Bowser's  Analytic  Geometry,  Taylor's  Calculus,  Church's  Descrip- 
tive Geometry,  Peck's  Determinants,  and  Todhunter's  Theory  of  Equa- 
tions. 

Cadet  Chester  Harding,  who  was  a  student  at  the  University  of 
Alabama  from  1881  to  1884,  gives  the  following  reminiscences  of  the 
mathematical  teaching  there  :  "  The  training  in  mathematics  was  more 
extensive  in  scope  and  thoroughness  in  the  engineering  course  than  in 
the  others,  including  in  that  course  the  elementary  principles  of  de- 
scriptive geometry  and  calculus,  while  in  the  others  the  instruction 
*  ceased  with  the  study  of  the  conic  sections  and  surfaces  of  the  second 
order  in  analytical  geometry. 

"  I  chose  the  engineering  coarse  and  began  my  instructions  in  the 
departmentof  mathematics  with  trigonometry  under  Prof:  W.  J.  Yaughn, 
who  now  fills  the  chair  of  mathematics  at  Vanderbilt  University.  Our 
text-book  was  Wheeler's  Trigonometry.  The  trigonometric  functions 
were  taught  as  ratios,  and  stress  was  laid  upon  the  circular  system  of 
measuring  angles.    *    *    * 

"  Analytical  geometry  came  next,  and  our  text-book  was  Professor 
Wood's  Elements  of  Co-ordinate  Geometry.  Of  the  class  of  thirty  in 
this  branch,  all  were  beginners  but  two  or  three  who  had  been  required 
to  repeat  the  course  because  of  their  deficiency  in  the  preceding  year. 
Our  progress  was  therefore  slow  at  first,  and  much  time  was  spent  by 
the  professor  in  explanations  and  illustrations.  I  see  the  first  lessons 
still  marked  in  the  text-book  I  have  before  me  now,  and  some  were  but 
two  and  a  half  of  the  quarto  pages.    These,  however,  were  expected  to 


INFLUX  OF  FRENCH  MATHEMATICS.  219 

be  thoroughly  mastered,  and  many  pains  were  taken  to  have  the  prin- 
ciples well  absorbed  by  the  students.  Mere  exercise  of  memory  was 
little  sought  after  in  the  mathematical  department,  and  any  originality 
on  the  part  of  a  student  in  the  Reduction  or  application  of  a  principle 
was  highly  commended. 

«'  The  course  in  analytical  geometry  closed  with  the  end  of  the  ses- 
sion, at  which  time  a  satisfactory  written  examination  in  the  study  was 
required  of  every  member  of  the  class.  In  the  scientific  and  classical 
courses,  mathematics  terminated  with  the  Sophomore  year.  In  the 
Junior  year  tbe  students  of  the  engineering  course,  however,  took  up 
the  study  of  calculus. 

«'  At  the  end  of  my  Sophomore  year  Professor  Vaughn  resigned  his 
chair  at  the  University  of  Alabama  to  accept  a  similar  position  at 
Vanderbilt.    *    *    * 

"  III  my  Junior  year  the  schedule  of  studies  was  so  arranged  that  but 
three  hours  a  week  were  devoted  by  my  class  to  mathematics.  This 
limited  time  permitted  us  to  complete  but  one  text-book.  Prof.  W.  G. 
Peck's  Elements  of  the  Differential  and  Integral  Calculus.  From  this* 
text,  however,  we  derived  a  knowledge  of  the  practical  utility  of  calcu- 
lus, and  became  familiarized  with  the  rules  of  diiferentiation  and  integra- 
tion. I  can  hardly  say  that  we  acquired  a  more  thorough  knowledge 
than  this ;  and  indeed  it  seemed,  from  the  time  assigned  to  the  study, 
to  be  without  the  purpose  of  the  faculty  that  more  than  a  groundwork 
should  be  acquired,  for  practical  good  in  the  understanding  of  the  ap- 
plications of  calculus  to  mechanics  and  engineering.  During  my  Jun- 
ior year  we  also  studied  under  Prof.  R.  A.  Hardaway,  in  the  depart- 
ment of  engineering,  the  elements  of  descriptive  geometry,  using  as  a 
text- book  Binn's  Elements  of  Orthographic  Projection. 

"With  the  close  of  the  Junior  year  the  regular  course  in  pure  mathe 
matics  was  ended." 

As  regards  the  conditions  for  graduation  which  have  existed  -at  vari* 
ous  times.  Professor  Palmer  says :  "  As  a  rule  mathematics  was  required 
of  every  student  for  graduation,  from  1831  to  1865.  After  the  reorgani- 
zation in  1869,  mathematics  was  also  required  until  1875,  when  the  elec- 
tive system  was  adopted ;  it  was  entirely  optional  with  the  student  then 
until  1880,  when  every  student  was  required  to  take  this  subject  through 
analytic  geometry." 

At  present  there  are  no  electives,  and  all  the  mathematics  in  each 
course  is  required  for  a  degree  in  that  course. 

UNIVERSITY  OF  MISSISSIPPI. 

The  educational  record  of  Mississippi  in  the  early  period  of  her  organ- 
ized existence  is  quite  honorable.  Between  1798  and  1848  there  had 
been  established  one  hundred  and  ten  institutions,  under  the  various 
names  of  universities,  colleges,  academies,  and  schools.  This  proves 
that  an  entire  obliviousness  to  the  educational  wants  of  the  people  did 


220  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

not  prevail.  Our  gratification  is  abated,  however,  by  the  consideration 
that  these  organizations  proved  inefficient,  and  tliat  there  was  really 
but  very  little  beneficial  progress. 

In  1848  was  organized  upon  a  firmer  foundation  the  University  of 
Mississippi.  Considering  the  many  difficulties  that  were  encountered, 
the  record  of  the  university  during  its  infant  years  before  the  War  was 
honorable.  Two  names,  both  well  known  to  the  educational  public, 
devoted  their  energies  to  promote  its  early  growth — F.  A.  P.  Barnard, 
now  president  of  Columbia  College,  and  A.  T.  Bledsoe,  afterward  pro- 
fessor at  the  University  of  Yirginia  and,  still  later,  editor  of  the  South- 
ern Methodist  Eeview. 

From  the  beginning  until  1854,  Albert  Taylor  Bledsoe  was  professor 
of  pure  and  applied  mathematics,  and  astronomy.  The  mathematical 
requirements  for  admission  were,  at  first,  a  knowledge  of  arithmetic. 
The  catalogue  of  1857-58  says  :  "Arithmetic — especially  the  subject  of 
fractions,  vulgar  and  decimal,  proportion,  and  the  extraction  of  roots ;" 
the  catalogue  for  1859-60  adds  to  this,  "  algebra  as  far  as  simple  equa- 
tions." In  the  former  catalogue  we  read  also,  "  that,  hereafter,  no 
student  will  be  admitted  to  any  class  in  the  university  who  shall  fail  to 
pass  an  entirely  satisfactory  examination  on  the  subjects  or  authors  re- 
quired for  admission  to  the  class." 

According  to  the  catalogue  of  1854,  the  Freshmen  studied  Davies' 
University  Arithmetic,  Davies'  Bourdon,  and  Davies'  Legendre ;  the 
Sopliomores  continued  Davies'  Bourdon  and  Legendre,  and  then  took  up 
Plane  and  Spherical  Trigonometry  and  Surveying ;  the  Juniors  studied 
Descriptive  Geometry,  Shades  and  Shadows  and  Perspective,  Davies' 
Analytical  Geometry,  and  Descriptive  Astronomy;  the  Seniors,  Davies' 
Differential  and  Integral  Calculus,  and  physical  astronomy.  In  the 
introduction  into  the  course  of  descriptive  geometry,  in  the  use  through- 
out of  Davies'  textbooks,  and  in  the  apparent  thoroughness  (for  that 
time)  of  the  mathematical  course,  we  observe  the  influence  of  the  U.  S. 
Military  Academy,  through  Professor  Bledsoe,  a  West  Point  graduate. 

When  Bledsoe  resigned  to  accept  a  professorship  at  the  University 
of  Yirginia,  Frederick  Augustus  Porter  Barnard,  a  young  man  of  re- 
markable mathematical  talents,  took  his  place.  Barnard  was  a  native  of 
Massachusetts  and  entered  Yale  college  in  1824.  Before  admittance  to 
college  he  had  given  no  time  to  mathematical  study  beyond  the  ele- 
ments of  arithmetic,  but  in  college  he  began  to  exhibit  decided  mathe- 
matical talent  and  taste.  His  tutor,  W.  H.  Holland,  later  professor  of 
mathematics  in  Trinity  College,  Hartford,  said  of  him  :  "  I  have  never 
known  any  person  except  the  late  lamented  Professor  Fisher,  who  pos- 
sessed so  extraordinary  natural  aptitude."  After  graduation  he  was, 
for  a  time,  tutor  at  Yale,  then  professor  at  the  University  of  Alabama, 
and,  in  1854,  became  Bledsoe's  successor  at  the  University  of  Missis- 
sippi. At  the  meeting  of  the  board  of  trustees,  in  July,  1856,  the  chair 
of  pure  and  applied  mathematics  and  astronomy  was  divided  into  two, 


INFLUX  OF  FRENCH  MATHEMATICS.  221 

the  chair  of  pure  mathematics,  and  the  chair  of  natural  science,  civil 
engineering,  and  astronomy.  Professor  Barnard  held  the  latter,  though 
he  continued  to  exercise  supervision  over  the  former,  and  was  also 
elected  president  of  the  university.  He  filled  these  ofiiees  until  the 
suspension  of  the  exercises  of  the  university,  in  1861. 

From  1856  to  1861  Jordan  McCullogh  Phipps  was  teacher  of  mathe- 
matics— at  first  adjunct  professor,  afterward  full  professor.  Daniel  B. 
Carr  was  tutor.  The  department  of  mathematics,  physics,  and  engi- 
neering seems  to  have  been  the  strongest  at  the  institution.  In  conse- 
quence of  frequent  complaint  that  the  general  statement  previously 
presented  in  the  annual  catalogues  of  the  university  had  been  unsat- 
isfactory, a  complete  account  of  expenses  and  of  the  courses  of  instruc- 
tion was  given  in  the  catalogues  issued  at  this  time.  From  the  one  of 
1857-58  we  quote  the  following: 

"  Instruction  in  pure  mathematics  commences  with  the  beginning  of 
the  Freshman  year,  and  is  continued  till  the  close  of  the  Sophomore. 
In  order  to  secure  greater  efficiency  of  instruction,  the  class  will  be  di- 
vided into  sections,  which  will  be  met  by  the  instructor  separately ; 
and  all  operations  in  this  and  every  other  branch  of  mathematical  sci- 
ence will  be  actually  performed  by  the  student  in  his  presence,  upon 
large  wall-slates  or  blackboards.  The  instructor  will  also  avail  himself 
of  the  same  means  of  illustrating  processes,  or  principles,  and  explain- 
ing difficulties. 

"  The  first  subject  attended  to  is  algebra.  It  will  be  the  instructor's 
endeavor  to  secure  a  thorough  acquaintance  with  the  elementary  prin- 
ciples of  the  science,  and  a  perfect  familiarity  with  its  practical  opera- 
tions. The  subject  effractions  will  be  especially  dwelt  on,  after  which 
will  follow  the  resolution  of  simple  equations,  numerical  and  literal,  in- 
volving one  or  more  unknown  quantities.  In  taking  up,  next  in  order, 
quadratic  equations,  the  first  object  will  be  to  secure  on  the  part  of  the 
student  a  perfect  understanding  of  the  form  of  the  binomial  square ; 
and  this  will  be  afterward  applied  to  the  completion  of  imperfect 
squares,  in  the  several  cases  in  which  one  of  the  terms  of  the  root  is  a 
number,  or  a  letter,  or  a  numerical  or  literal  fraction.  The  method  be- 
ing generalized,  will  then  be  applied  to  the  reduction  of  abstract  equa- 
tions, and  the  statement  and  resolution  of  problems  involving  quad- 
ratics. Where  the  equation  is  denominate,  the  student,  will  be  re- 
quired to  interpret  the  result,  to  explain  the  ambiguous  sign,  and  to 
distinguish  cases  in  which  the  conditions  of  the  problem  involve  an  im- 
possibility.    *    *    * 

"  The  subject  of  algebra  will  be  completed  by  the  discussion  of  the 
general  theory  of  equations,  their  formation,  their  solution,  and  their 
properties,  including  in  the  course  the  ingenious  theorem  of  Sturm. 

"  In  all  parts  of  this  subject,  encouragement  will  be  held  out  to  stu- 
dents to  exercise  their  ingenuity  in  devising  various  modes  of  arriving 
at  the  same  results ;  and  special  merit  will  be  attached  to  the  processes 


222  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

wliich  are  the  most  succinct  or  elegatit.  As  a  stimulus  to  this  species 
of  iugeimity,  problems  not  embraced  in  the  text-book  may  from  time  to 
time  be  proposed  by  the  instructor ;  and  varieties  in  the  mode  of  state- 
ment both  of  these  and  of  those  which  occur  in  the  regular  course,  will 
be  called  for  from  any  who  may  choose  to  present  them* 

"  Geometry,  plane,  solid,  and  spherical,  will  occupy  the  latter  portion 
of  the  Freshman  year.  In  this  branch  of  science  all  demonstrations  will 
be  made  from  figures  drawn  upon  the  blackboards,  or  wall-slates,  by 
the  student  reciting,  and  promptness  and  accuracy  in  this  part  of  the 
business  will  be  urgently  inculcated  and  regarded  as  a  merit.  The 
Student  will,  moreover,  be  advised  to  avoid  a  servile  imitation  of  the 
exact  forms  of  the  diagrams  given  in  the  text-book,  and  will  have  his 
ingenuity  exercised  either  in  forming  other  figures  to  illustrate  the  same 
propositions,  or  in  demonstrating  the  propositions  from  figures  con- 
structed for  him.  He  will  also  be  required  to  adopt  a  mode  of  lettering 
his  figures  different  from  that  of  the  book;  or  to  give  the  demonstrations 
without  the  use  of  letters  at  ail,  by  pointing  to  the  parts  of  the  figure 
successively  referred  to  in  the  demonstration. 

"  It  v.^ill  always  be  regarded  as  specially  meritorious  in  a  student  to 
present  a  demonstration  of  any  proposition  founded  on  any  legitimate 
method  diiferiug  from  that  of  the  author ;  and  the  instructor  will,  him- 
self, from  time  to  time,  illustrate  this  practice,  by  way  of  awakening  the 
ingenuity  of  the  student.  For  the  purpose  of  still  further-  encouraging 
originality  of  investigation,  and  exciting  honorable  emulation,  the  plan 
already  described  as  to  be  pursued  in  algebra,  will  be  continued  here, 
of  propounding  propositions  not  contained  in  the  text-book,  of  which 
demonstrations  will  be  subsequently  called  for,  and  which  will  secure 
special  distinction  to  such  as  satisfactorily  solve  them." 

Equally  full  is  the  account  of  the  mathematical  work  in  the  Sopho- 
more year.  The  studies  for  that  year  were  plane  and  spherical  trigo- 
nometry, mensuration,  surveying,  leveling,  navigation,  and  analytical 
geometry.  Considerable  field-work  was  done  in  surveying.  The  lev- 
eling rods  employed  had  the  common  division  to  feet  and  fractions,  and 
also  the  French  metrical  division. 

The  catalogue  then  proceeds  as  follows : 

"  The  course  of  pure  mathematics  will  conclude  with  the  subject  of 
the  differential  and  integral  calculus,  which  will  be  taught  at  the  end 
of  the  Sophomore  or  the  beginning  of  the  Junior  year.  This  will  em- 
brace the  doctrine  of  functions,  algebraic  and  transcendental,  the  dif- 
ferentiation of  functions,  successive  differentials,  theorems  of  Taylor 
and  Maclaurin,  logarithmic  series,  the  development  of  a  circular  arc  in 
terms  of  its  functions,  or  of  the  functions  in  terms  of  the  arc,  partial 
differentials,  differential  equations  of  curves,  principles  of  maxima  and 
minima,  expressions  for  tangents  and  normals,  singular  and  multiple 
points,  osculating  circles,  involutes  and  evolutes,  transcendental  curves, 
and  spirals  ;  the  integration  of  regularly  formed  differentials,  Integra- 


INFLUX  OF  FEENCH  MATHEMATICS.  223 

f ion  by  serieSj  integration  of  rational  and  irrational  fractions,  special 
methods  of  integration,  the  rectification  of  curves,  the  quadrature  of 
curves  and  curved  surfaces,  the  cubature  of  solids,  and  the  integration 
of  differentials  of  two  or  more  variables." 

In  natural  philosophy  great  efforts  were  made  to  secure  a  complete 
set  of  apparatus.  In  the  catalogue  for  1857-58  we  read  as  follows : 
"  It  is  probable  that,  with  the  opening  of  the  ensuing  session,  the  elec- 
trical apparatus  of  the  University  of  Mississippi  will  be  superior  to  any 
similar  collection  in  the  United  States.'^ 

In  astronomy  the  celestial  motions  were  beautifully  represented  by 
Barlow's  magnificent  planetarium,  eleven  feet  in  diameter — ^"apiece 
of  mechanism  unrivaled  in  ingenuity,  accuracy,  and  elegance."  A  port- 
able transit  instrument  was  also  available  for  observations  of  meridian 
passages,  and  a  sextant  and  a  prismatic  reflecting  circle  furnished  means 
of  making  direct  measurements  of  altitudes  and  arcs.  The  catalogue 
then  says: 

"  The  course  of  civil  engineering,  distinctly  so  called,  falls  entirely 
within  the  Senior  year ;  but  it  is  in  considerable  part  only  a  further 
development  and  application  of  principles  embraced  in  the  sciences  of 
pure  mathematics  and  physics  previously  taught.  The  course  will  em- 
brace geometrical  and  topographical  drawing,  the  use  of  field  instru- 
ments, such  as  the  engineer's  transit,  the  goniasmometer  or  pantometer, 
the  leveling  instrument,  the  theodolite,  the  sextant,  the  reflecting  cir- 
cle, and  the  plane  table,  descriptive  geometry,  trigonometrical  survey- 
ing and  geodesy,  marine  surveys,  materials  of  structures,  engineering 
statics,  carpentry,  masonry,  bridge  construction,  surveys  for  location 
and  construction  of  roads  and  railroads,  laying  out  curves,  staking  out 
cuts  and  fills,  hydraulic  engineering,  drainage,  canals,  locks,  aqueducts, 
dams,  sea  walls,  river  improvements,  and  the  dynamics  and  economy  of 
transportation.    *    *    * 

"  Throughout  every  part  of  the  course,  the  student  will  be  constantly 
encouraged  and  stimulated  to  consult  other  authorities  on  the  subjects 
taught,  besides  the  text-books  :  and  the  instructors  will  often  refer  them 
on  special  subjects,  to  such  authorities.  The  following  list  embraces 
the  text-books  (first  in  order),  and  the  authors  to  whom  reference  will 
most  frequently  be  made : 

"Algebra:  Perkins,  Hackley,  Peirce. 

Geometry  :  Perkins,  Playfair  (Euclid),  Peirce. 

Tkigonometrt  :  Perkins,  Hackley,  Peirce. 

Surveying  :  Gillespie,  Davies,  Gummere. 

Analytical  GeoiiIETRy  :  Davies,  Peirce. 

Calculus  :  Davies,  Peirce,  Cliurch,  Jephson. 

Natural   Philosophy:  Olmsted,  Bartlett,  Whewell,  Brewster  ("Optics), 
Herschel  (Light  and  Sound"),  Peirce. 

Astronomy  :  Olmsted,  Gummere,  Bartlett,  Loomis. 

Civil  Engineering  :  Malian,  Moseley,  'Wiesbacii,  Gillespie,  Haupt,  Bourne, 
Pambouf." 


224  TEACHING   AND    HISTOKY    OF   MATHEMATICS. 

The  courses  for  the  remainini?  years  before  the  War  were  essentially 
the  same  as  the  one  we  have  described.  The  constant  use  of  the  black- 
board is  emphasized  throughout.  The  fact  that  pains  are  taken  to 
explain  the  term  as  meaning  ''  large  wall-slates"  rather  tends  to  show 
that  blackboards  were  then  a  novelty  in  Mississippi.  As  far  as  we  can 
judge  from  the  catalogues,  the  instruction  was  methodical  and  of  high 
efficiency.  A  serious  drawback  to  high  scholarship  was  found,  no 
doubt,  in  the  lack  of  preliminary  culture  and  training  in  students  enter- 
ing the  university. 

The  attendance  of  students  was  good.  The  numbe?  of  graduates  from 
the  department  of  arts  from  1851  to  1859,  inclusive,  was  268.  During 
the  last  year  before  the  War  the  number  of  students  in  the  college  was 
191,  of  whom  16  were  "  irregular"  in  grade. 

Owing  to  the  universal  enlistment  of  males,  even  youths,  in  the  Con- 
federate States  army,  the  university  exercises  were  suspended  in  1861, 
until  October  1865.  In  1865  General  Claudius  W.  Sears,  ex-brigadier- 
general  of  the  Confederate  States  army,  and  a  graduate  of  West  Point, 
was  elected  professor  of  mathematics.    This  position  he  still  holds. 

The  mathematical  requirements  for  entering  were,  in  1866,  "arithme- 
tic and  algebra,  including  equations  of  the  first  degree."  The  course 
of  pure  mathematics  for  the  regular  under-graduate  curriculum  was 
completed  at  the  end  of  the  Sophomore  year,  and  consisted  of  Bourdon's 
Algebra,  Legendre's  Geometry,  Trigonometry,  Mensuration,  Surveying, 
and  Analytical  Geometry. 

A  more  extended  course  than  was  required  for  the  degree  of  bachelor 
of  arts  could  be  obtained  in  the  department  of  applied  mathematics  and 
civil  engineering,  which  was  in  charge  of  General  F.  A.  Shoup,  a  grad- 
uate of  West  Point,  and  now  of  the  University  of  the  South.  The 
course  of  instruction  in  his  department  formed  no  necessary  part  of  the 
under-graduate  course.  It  was  designed  to  meet  the  wants  of  such 
students  as  intended  to  make  civil  engineering  or  some  other  of  the 
mechanic  arts  a  profession.  In  this  course  analytical  geometry  and 
calculus  were,  of  course,  indispensable,  and  they  could  be  studied 
while  students  were  pursuing  their  branches  in  the  department  proper. 
The  course  could  be  completed  by  an  ordinary  student  who  came  fairly 
well  prepared  in  preliminary  branches  in  about  two  years. 

In  1870  the  plan  of  instruction  in  the  university  was  altered  so  as  to 
include  (1)  a  department  of  preparatory  education,  (2)  a  department  of 
science,  literature,  and  arts  (leading,  respectively,  to  the  degrees  of  B. 
A.,  B.  S.,  B.  Ph.,  0.  E.),  and  (3)  a  department  of  professional  education 
(law). 

The  terms  for  admission  into  the  bachelor  of  arts  and  bachelor  of 
science  courses  were,  in  mathematics,  arithmetic,  and  Davies'  Ele- 
mentary Algebra  through  equations  of  the  second  degree.  Candidates 
for  the  bachelor  of  philosophy  course  and  civil  engineering  were  exam-' 
ined  on  the  whole  of  Davies'  Elementary  Algebra.     These  requirements 


INFLUX  OF  FRENCH  MATHEMATICS.  225 

have  remained  unchanged  till  the  present  time.  The  department  of 
civil  engineering  was  discontinued  in  1876.  In  1872  the  first  year's 
mathematical  work  in  the  course  leading  to  the  B.  A.,  B.  S..  and  B.  Ph. 
degrees  consisted  in  the  study  of  Davies'  Bourdon's  Algebra,  and  Le- 
gendre's  Geometry  and  Plane  Trigonometry.  During  the  first  half  of 
the  Sophomore  year  Church's  Analytical  Geometry  and  Davies'  Land 
Surveying  (with  use  of  instruments  in  the  field)  were  studied.  Tliis 
completed  the  course  in  pure  mathematics.  A.  B.  students  were  taught 
Smith's  Mechanics  and  Hydrostatics,  Hydraulics,  and  Sound  (Bartlett) 
in  the  Junior  year,  and  Bartlett's  Optics  and  Astronomy  in  the  Senior 
year.  B.  S.  students  had  Gummere's  Astronomy  in  the  second  half  of 
the  third  year.  (The  B.  S.  and  B.  Ph.  were  the  only  three  years' 
courses.) 

At  the  present  time  (1888)  the  mathematical  course  is  decidedly 
stronger.  Van  Amringe's  edition  of  Davies'  text-books  are  used,  ex- 
cept in  analytical  geometry  and  calculus,  which  are  studied  from  the 
works  of  Church.  The  calculus  is  now  studied  during  the  latter  part 
of  the  Sophomore  year. 

Prof.  C.  W.  Sears  has  now  occupied  the  mathematical  chair  for 
twenty-three  years.  One  of  his  old  pupils,  Prof.  Edward  Mayes,  says 
of  him,  "that  he  'quizzes'  'like  all  possessed,'  pretends  that  he  does 
not  know  anything  about  it,  and  asks  'all  sorts  of  impertinent  ques- 
tions.'" As  Sydney  Smith  said  of  Alexander  Pope,  "I  studied  under 
him,  and  have  lively  recollections." 

KENTUCKY  UNIVEKSITY. 

The  records  of  the  Transylvania  University  for  several  years  follow- 
ing 1817  appear  to  have  been  lost.  In  1825  Thomas  J.  Matthews,  the 
father  of  the  late  Justice  Stanley  Matthews,  of  the  Supreme  Court  of  the 
United  States,  is  mentioned  as  being  "professor  of  mathematics  and 
natural  philosophy."  The  subjects  taught  by  him  were  "arithmetic, 
geometry,  surveying,  leveling,  natural  philosophy,  and  book-keeping." 
The  entry  for  1829  shows  that  Pestalozzian  ideas  had  gained  a  foothold 
at  the  university,  inasmuch  as  Col  burn's  Algebra  is  mentioned  as  the 
mathematical  text-book  for  the  Freshmen.  The  SopJwmores  studied 
Playfair's  Geometry  and  Trigonometry;  tliQ  Juniors,  Day's  I^avigation, 
Surveying,  Heights  and  Distances,  Leveling;  the  Seniors,  Bezout's 
Fluxions.  Bezout's  text-book  had  been  translated  from  the  French  by 
Professor  Farrar,  of  Harvard.  It  employed  the  notation  of  Leibnitz, 
and  did  not  therefore  teacli  "fluxions."  The  use  of  this  term  as  a 
synonym  for  "differential  and  integral  calculus"  was,  we  believe,  pecu- 
liarly American. 

In  1832  John  Lutz  was  elected  "  professor  of  mathematics  and  natural 
philosophy, "  and  in  1837  Benjamin  Moore.  The  latter  resigned  after 
one  year's  service. 

881— Ko.  3 15 


226  TEACHING   AND    HISTOEY    OF    MATHEMATICS. 

The  records  from  1839  to  1865  can  not  be  found.  From  old  catalogues 
we  glean  the  following  :  In  1844  E.  T.  P.  Allen  was  professor,  and  the 
subjects  taught  were,  in  the  Freshman  year,  Davies'  Bourdon  and  Le- 
gendre  ;  in  the  So^pJiomore  year,  plane  and  spherical  trigonometry, 
heights  and  distances,  mensuration  of  superficies  and  solids  (Davies'), 
navigation  (Day's),  conic  sections  (Davies'  Analytical  Geometry),  Sur- 
veying (Davies'),  descriptive  geometry  (Davies') ;  in  the  Junior  year, 
differential  and  Integral  calculus  (Davies') ;  in  the  Senior  year,  Olm- 
sted's Astronomy. 

In  1848  James  B.  Dodd  held  the  chair  of  mathematics  and  natural 
philosophy.  At  this  time  the  course  was  as  follows  :  Freshman  year, 
arithmetic  reviewed,  Loomis's  Algebra,  five  books  of  Legendre ;  Sopho- 
more year,  geometry  completed,  plane  and  spherical  trigonometry  and 
their  applications,  analytical  geometry  (Davies',  6  books) ;  Junior  year, 
Church's  Calculus. 

In  1850  the  mathematics  for  the  Junior  and  Senior  classes  consisted 
of  descriptive  geometry,  analytical  geometry,  calculus,  and  analytical 
mechanics ;  but  they  were  optional  with  the  student. 

Prof.  James  B.  Dodd  was  the  most  prominent  mathematical  teacher 
that  was  connected  with  Transylvania  University.  He  was  a  native  of 
Yirginia,  and  a  self-made  mathematician.  In  1841  he  became  professor 
of  mathematics  at  the  Centenary  College  in  Mississippi,  and  in  1846 
was  elected  professor  at  the  Transylvania  University.  He  published 
several  books,  viz.,  an  Elementary  and  Practical  Arithmetic,  High  School 
Arithmetic,  Elementary  and  Practical  Algebra,  Algebra  for  High 
Schools  and  Colleges,  and  Elements  of  Geometry  and  Mensuration. 
Some  of  these  reached  several  editions.  Professor  Dodd  contributed 
also  to  the  Quarterly  Eeview  of  the  M.  E.  Church  South.  In  1849  he 
was  appointed  president  pro  tempore  of  the  university. 

In  1865  Transylvania  University  was  merged  into  Kentucky  Univer- 
sity. The  chair  of  mathematics  in  Kentucky  University  has  been  filled 
from  1859  to  the  present  time  by  Henry  H.  White.  From  1870  to  1876 
James  G.  White  acted  as  adjunct  professor.  From  1876  to  1878  he  was 
professor.  In  mathematics  the  requirement  for  admission  has  been 
algebra  through  equations  of  the  first  degree.  When  Prof.  Henry  H. 
White  first  became  connected  with  the  university  as  professor,  the 
course  was  as  follows :  Algebra  completed,  plane  and  solid  geometry, 
application  of  algebra  to  geometry,  plane  and  spherical  trigonometry, 
surveying  and  navigation,  analytical  geometry,  differential  and  integral 
calculus,  mechanics,  and  astronomy,  witk  original  iDroblems  and  exer- 
cises throughout  the  course  when  practicable.  In  1864  the  course  was 
modified  by  dropping  applications  of  algebra  to  geometry;  in  1879,  by 
the  addition  of  conic  sections  (treated  geometrically) ;  and  in  X884,  by 
dropping  conic  sections  and  navigation. 

The  text-books  used  by  Prof.  Henry  H.  White  at  different  times  are 
as  follows:  In  Algebra,  Davies'  Bourdon,  Towne,  Peckj  in  Geometry^ 


INFLUX  OF  FRENCH  MATHEMATICS.  227 

Davies'  Legendre,  Peck;  in  Trigonometry,  Davies,  Peck;  in  Surveying 
and  Navigation,  Davies,  Loomis  5  in  Analytical  Geometry,  Loomis,  Peck ; 
in  Calculus,  Loomis,  Peck;  in  Mechanics,  Olmsted,  Snell's  Olmsted, 
Peck ;  in  Astronomy,  Olmsted,  SnelPs  Olmsted,  Peck. 

There  have  been  no  electives  in  mathematics  up  to  this  time,  except 
that  the  student  now  has  the  choice  between  languages  and  calculus. 

UNIVEESITY  OF   TENNESSEE.* 

"  The  foundation  of  this  university  is  connected  with  the  earliest 
history  of  Tennessee. 

"  In  1794,  by  the  first  General  Assembly  of  the  '  Territory  south  of  the 
Ohio,'  was  chartered  Blount  College,  named  in  honor  of  William  Blount, 
Governor  of  the  Territory,  and  afterward  one  of  the  two  United  States 
Senators  first  chosen  from  the  State  of  Tennessee. 

"  In  1807,  under  an  act  of  Congress  providing  for  the  establishment 
of  two  colleges  in  Tennessee,  East  Teauessee  College  was  chartered, 
and  soon  after  the  franchise  and  property  of  Blount  College  were  trans- 
ferred to  the  new  institution.    *    *    * 

"  In  1840  the  name  of  East  Tennessee  College  was  changed,  by  act 
of  Legislature,  to  Bast  Tennessee  University. 

''  Iq  1869  the  Legislature  gave  in  trust  to  the  university  the  pro- 
ceeds of  the  sale  of  public  lands,  donated  by  act  of  Congress  of  July  2, 
1862,  '  to  the  several  States  and  Territories  which  may  provide  col- 
leges for  the  benefit  of  agriculture  and  the  mechanic  arts.' 

"In  1879  the  name  of  East  Tennessee  University  was  changed,  by  an 
act  of  the  Legislature,  to  the  University  of  Tennessee."  t 

It  is  a  source  of  regret  to  us  that  we  have  not  been  able  to  obtain 
any  information  whatever  on  the  mathematical  instruction  at  this  in- 
stitution during  the  first  eighty  years  of  its  existence.  Ever  since  it 
took  the  name  of  a  university,  it  has  been  in  an  almost  continual  state 
of  reorganization.  These  constant  upheavals  have  resulted  in  the  loss 
of  almost  all  its  records.  "The  requirements  for  admission  and  grad- 
uation," says  Professor  Carson,  "  have  probably  been  changed,  on  an 
average,  every  two  years."  The  terms  for  admission  were  not  rigidly 
adhered  to,  and  the  standard  for  graduation  has  not  always  been  high. 

The  catalogue  of  1874-75,  the  earliest  one  that  we  have,  gives  John 
Kerr  Payne  as  professor  of  mathematics  and  mechanical  philosophy. 
The  collegiate  department  comprised  at  this  time  three  distinct  courses 
viz.,  the  agricultural  course,  the  mechanical  course,  and  the  classical 
course.  The  standard  for  admission  to  the  first  two  courses  was, 
until  1874,  lower  than  to  the  last  course.  In  1874-75  the  mathematical 
studies  in  the  agricultural  course  were  according  to  catalogue,  as  fol- 

*  The  writer  ia,  indebted  to  Prof.  Wtn.  W.  Carson,  professor  of  matlieaiatios  and  civil 
engineering  at  the  University  of  Tennessee,  for  all  the  iaformatioa  herein  coutaiuede 
t  Catalogue  of  the  University  of  Tennessee,  1885-86. 


228  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

lows:  Freshmen,  Eobinson's  University  Algebra,  beginning  with  quad- 
ratic equations,  Chauvenet's  Geometry,  beginning  at  tlie  third  book, 
Loomis's  Conic  Sections;  Sophomores,  Church's  Descriptive  Geometry, 
Loomis's  Trigonometry  and  Surveying;  Juniors,  Olmsted's  Natural 
Philosophy  and  Astronomy.  In  the  mechanical  and  classical  courses, 
the  schedule  was  the  same  in  mathematics,  except  that  spherical  trigo 
nometry,  Loomis's  Analytical  Geometry  and  Calculus,  and  civil  engi- 
neering, were  added. 

The  biennial  report  of  the  trustees  for  1881  gives  James  Dinwiddie  as 
professor  of  pure  mathematics,  and  Samuel  H.  Lockett  as  professor  of 
applied  mathematics  and  mechanical  philosophy.  The  report  shows  that 
the  university  was  then  organized  into  distinct  schools,  like  the  Uni- 
versity of  Virginia.  These  schools  have  existed,  probably,  since  1879. 
Of  the  school  of  pure  mathematics,  the  report  says : 

"  The  subjects  taught  in  the  subcollegiate  year  of  this  school  are  ele- 
mentary algebra,  and  four  books  of  geometry.  In  the  first  collegiate 
year  algebra  and  geometry  are  finished,  and  plane  trigonometry  is 
studied.  In  the  second  collegiate  year  are  studied  spherical  trigonom- 
etry and  analytical  geometry  of  two  dimensions,  and  in  the  third  year 
differential  and  integral  calculus." 

The  work  in  the  school  of  applied  mathematics  is  described  as  fol- 
lows : 

"  Elementary  experimental  physics  is  taught  in  the  first  college  year. 
The  various  subjects  of  statics  and  dynamics  of  solids,  liquids,  and 
gases;  of  acoustics,  heat,  light,  electricity,  and  magnetism,  are  treated 
without  the  aid  of  the  mathematics,  and  are  illustrated  by  numerous 
experiments.  The  apparatus  has  been  specially  selected  for  that  pur- 
pose. 

"In  the  analytical  mechanics,  the  power  of  the  whole  range  of  the 
mathematics  is  brought  to  bear  upon  the  investigation  of  the  laws  of 
forces  of  nature,  and  the  student  is  made  familiar  with  the  power  and 
utility  of  mathematics  by  the  solution  of  a  large  number  of  practical 
problems.  Astronomy  has  thus  far  been  taught  without  instruments, 
but  the  board  of  trustees  has  apiDropriated  five  hundred  dollars  for  the 
purchase  of  a  telescope.  Surveying  comprehends  plane  surveying,  lev- 
eling, topographical  surveying,  and  mining  surveying;  the  use  of  the 
compass,  transit,  Y  level,  plane  table,  chain,  and  leveling  rod;  also 
plotting,  making  profiles  and  cross-sections,  and  topographical  drawing 
with  pen  and  brush.  A  large  share  of  the  student's  time  is  given  to 
field  work  and  practice. 

"Descriptive  geometry  is  the  foundation  of  both  the  science  and  art 
of  drawing.  It  is  followed  by  a  course  of  problems  in  shades,  shadows, 
and  i)erspective — mechanical  drawing. 

"The  course  of  engineering  consists  of  the  subjects  treated  in  Pro- 
fessor Gillespie's  Eoads  audEailroads  and  Professor  Wood's  revision  ot 
Mahan's  Civil  Engineering,  and  of  a  course  of  lectures  by  the  instructor 


INFLUX  OF  FRENCH  MATHEMATICS.  229 

on  surface  and  tliorough  drainage,  on  agricultural,  hydraulic,  and  ma- 
rine engineering,  and  a  brief  outline  of  the  science  and  art  of  military 
engineering.  The  engineering  drawing  consists  of  a  course  of  instruc- 
tion in  the  drawing  of  plans,  sections,  elevations,  and  details  of  bridges, 
tunnels,  canal  locks,  etc. 

"  For  the  above  engineering  course  students  can  substitute  mechan- 
ism, machinery,  and  machine  drawing." 

The  catalogue  for  1883-84  mentions  as  text-books  in  the  school  of 
pure  mathematics:  '•'White's  or  OIney's  Arithmetic;  Davies'  Bour- 
don, or  OIney's  Algebra ;  OIney's  Trigonometry ;  Bowser's  or  Peck's 
Analytical  Geometry;  Bowser's  or  Peck's  Calculus;  Bledsoe's  Philoso- 
phy of  Mathematics. 

"  Extra  examples,  illustrating  the  different'  subjects  taught,  are  given 
throughout  the  course." 

This  is  the  first  time  that  we  find  Bledsoe's  Philosophy  of  Mathe- 
matics named  as  one  of  the  text-books  in  a  college  course.  According  to 
catalogue,  it  was  used  in  the  third  collegiate  class,  which  completed 
analytic  geometry  and  then  took  up  "differential  and  integral  calculus, 
and  the  philosophy  of  mathematics."  The  idea  of  teaching  the  philos- 
ophy of  mathematics  is  certainly  a  good  one,  but  the  subject  is  hardly 
presented  by  Bledsoe  in  a  form  suitable  for  a  young  student. 

In  the  school  of  applied  mathematics  the  books  given  in  the  cata- 
logue for  1883-84  are.  Gage's  Physics ;  Loomis's  Ajstronomy ;  Davies' 
Xew  Surveying ;  Smith's  Topographical  Drawing ;  Church's  Descrip- 
tive Geometry ;  Wood's,  or  Eankine's  Mechanics ;  Mahan's  Civil  En- 
gineering ;  Searles's  Field  Engineering. 

In  June,  1888,  a  reorganization  and  a  re-classiflcation  of  the  various 
schools  took  place.  The  work  of  the  "  school  of  mathematics  and  civil 
engineering  "  for  the  year  1888-89  is  as  follows : 

I.   MATHEMATICS. 

First  c7ass— (Sab-FresTiman) :  Algebra  (through  surds  and  quadratics) ;  Geometry 
(three  books). 
Second  class — (Freshman) :  Geometry,  Algebra. 

Third  cZass— (Sophomore) :  Trigonometry  ;  Graphic  Algebra  ;  Analytical  Geometry. 
Fourth  class — (Junior)  :  Calculus. 

Each  class  is  taught  iu  sections  small  enough  to  be  well  handled  by 
the  instructor.  Great  stress  is  laid,  throughout  the  course,  on  the 
written  solution  of  original  problems — ^the  aim  being  to  induce  clear- 
ness of  thought  by  precision  in  expression.  Each  student  is  required 
to  use  the  level,  transit,  and  compass,  from  the  beginning  of  his  Fresh- 
man to  the  end  of  his  Sophomore  year.  On  entering  the  Freshman 
class  the  use  and  adjustments  of  the  level  are  explained  to  him.  He 
then  practices  with  it,  at  times  convenient  to  himself,  until,  by  running 
such  lines  as  may  be  required  of  him  and  submitting  profiles  and  cross- 
sections,  he  shows  his  ability  to  handle  the  ordinary  problems  of  drain- 


230  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

age  and  irrigation.  The  graphical  problems  in  geometry  are  solved, 
sometimes  with  drawing  instruments  on  paper,  and  sometimes  with 
engineering  iiistrameats  on  the  ground.  Thus  habits  of  accuracy  are 
enforced  early  in  the  course  by  the  use  of  instruments  of  precision,  and 
an  elementary  knowledge  of  surveying  afforded. 

For  admission  to  the  first  class  the  applicant  is  examined  in  arith- 
metic only. 

The  text-books  now  in  use  are  as  follows:  Hall  and  Knight's  Algebra 
for  the  Sub-Freshman  class,  Wentworth's  Algebra  for  the  Freshman 
class,  Wentworth's  Geometry,  Wells's  Trigonometry,  Puckle's  Oouic 
Sections  (with  lectures),  Newcomb's  Calculus.  The  Calculus  is  taught 
mainly  by  lectures,  the  text-book  being  used  as  a  guide.  As  taught  at 
present,  it  is  based  on  the  Idea  of  fluxions,  demonstrated  by  limits,  and 
employs  the  notation  of  Leibnitz.  In  pure  mathematics  no  higher 
branches  than  the  calculus  have  been  taught  at  the  university,  except 
during  the  se^ion  l(S86-87,  when  a  class  in  quaternions  was  taught. 
At  present  agricultural  students  must  finish  trigonometry,  all  others 
analytical  geometry,  while  the  engineering  students  must  finish  calculus. 

II.   CIVIL  ENGIXEEEING. 

1.  (Sopliomore):  Descriptive  Geometry ;  Land,  City,  and  Mine  Surveying. 

2.  (Junior):  Stone  Cutting ;  Astronomy. 

3.  (Junior) :  Elementary  Mechanics  ;  Analytical  Mechanics. 

4.  (Junior):  Surveys;  Soundings;  Maps;  Profiles;  Cross-sections;  Estimates'; 
Laying  out  Work ;  Engineering  Materials  and  Methods, 

The  time  of  this  class  is  mainly  spent  in  practical  work.  It  makes 
barometric  reconnaissances ;  makes  a  map  of  some  portion  of  the  bed  of 
the  Tennessee  Eiver ;  does  the  field  and  office  engineering  work  for  a 
line  of  communications  to  join  two  selected  points,  etc. 

5.  (Senior):  Analytical  Mechanics ;  Applied  Mechanics. 

6.  (Senior) :  Engineering  Structures  ;  Specifications  and  Contracts. 

7.  (Post-graduate):  Economics  ofEoads;  Sewerage;  Water  Supply;  Hydraulics; 
Architecture, 

The  department  is  admirably  equipped  with  the  various  engineering 
instruments.  Of  the  mpre  important  (such  as  levels,  transits,  sextants, 
aneroids,  etc.)  it  has  a  number  of  each.  It  has,  witb  great  care  and  ex- 
pense, procured  instruments  of  the  finest  workmanship  and  latest  at- 
tachments, so  that  its  students  of  engineering  may  see  how  much  to 
expect  the  iustrument-maker  to  contribute  toward  the  attainment  of 
accuracy  and  speed.  Exercises  requiring  their  use  are  continually  re- 
quired of  every  class. 

The  first  six  of  these  classes  are  required  for  the  degree  of  bachelor  of 
science  in  civil  engineering — the  seven  for  the  degree  of  civil  engineer. 

At  present  the  University  of  Tennessee  is  entering  upon  a  career  of 
remarkable  prosperity.  Like  most  of  the  higher  institutions  of  learning 
in  the  South,  it  is  experiencing  a  great  revival.  More  thorough  work 
and  a  higher  standard  of  scholarship  are  everywhere  perceivable. 


INFLUX  OF  FRENCH  MATHEMATICS.  23l 

The  present  prosperity  of  the  University  of  Tennessee  is  due  chiefly 
to  the  aggressive  leadership  of  its  President,  Dr.  C.  W.  Dabney,  a  grad- 
uate of  the  University  of  Virginia,  and  later  of  the  University  of  Got- 
tingen.  Ee  accepted  the  presidency  in  August,  1887,  under  conditions 
giving  him  great  freedom  to  manage  the  institution  according  to  his 
own  ideas.  In  June,  1888,  the  professorships  were  declared  vacant, 
and  were  then  filled  by  men  selected  by  the  president.  Prof.  William 
W.  Carson,  who  had  been  elected  to  the  chair  of  mathematics  in  1885, 
was  now  elected  professor  of  mathematics  and  civil  engineering.  Pro- 
fessor Carson,  a  graduate  of  Washington  and  Lee,  was  civil  engineer 
for  a  number  of  years.  Of  the  other  teachers  of  pure  and  applied  math- 
ematics, Prof.  T.  F.  Burgdorff  served  about  a  dozen  years  in  the  U.  S. 
Navy,  and  Prof.  E.  E.  Gayle  about  an  equal  length  of  time  in  the  U.  S, 
Army.    The  three  other  instructors  in  this  school  are  young  men. 

TULANE  UNIVERSITY  OF  LOUISIANA. 

TheTulane  University  came  into  existence  as  such  in  1884,  when,  by 
a  contract  with  the  State  of  Louisiana,  the  administrators  of  the  Tulaue 
educational  fund  became  the  administrators  of  the  University  of  Lou- 
isiana in  perpetuity,  agreeing  to  devote  their  income  to  its  development. 

The  University  of  Louisianahad  its  origin  in  the  Medical  Department, 
which  was  established  in  1834.  This  school  has  numbered  among  its 
professors  and  alumni  the  most  distinguished  medical  men  of  Louisiana 
and  the  South.  A  law  department  was  organized  in  1847  ;  and  in  1878 
the  academic  department  of  the  University  of  Louisiana  was  opened. 
It  existed  under  that  name  till  1884,  when  it  was  absorbed  into  Tu- 
lane  University.  Considering  that  the  academic  department  of  the  Uni- 
versity of  Louisiana  received  from  the  State  an  annuity  of  only  ten  thou- 
sand dollars,  it  met  with  excellent  success.  A  number  of  very  earnest 
and  well-trained  young  men  were  graduated  during  the  six  years  of  its 
existence.  Its  faculty  consisted  of  only  seven  professors,  but  they  were 
men  of  energy  and  ability.  E.  H.  Jesse  was  dean  of  the  faculty  and 
professor  of  Latin.  He  was  educated  at  the  University  of  Virginia,  and 
was  a  man  of  unusual  executive  ability.  His  individuality  was  strongly 
felt  in  the  institution.  He  organized  the  department,  taking  the  Uni- 
versity of  Virginia  as  his  model.  There  was  no  curriculum  or  prescribed 
course  of  study.  The  parent  or  guardian  had  to  choose,  with  the  advice 
of  the  faculty,  the  branches  to  be  pursued  by  the  student.  His  cast  of 
mind,  as  well  as  his  future  vocation,  could  thus  receive  due  Weight.  In 
1883  there  were  eight  "  schools."  The  student  was  required  to  attend 
at  least  three,  but  he  was  discouraged  from  electing  more  than  four,  in 
order  to  prevent  superficial  work. 

The  school  of  mathematics  was  in  charge  of  J.  L.  Cross,  the  profes- 
sor of  mathematics.  Professor  Cross  was,  before  the  War,  a  student 
at  the  Virginia  Military  Institute,  and  a  pupil  of  Prof.  Francis  H. 
Smith.    The  school  of  mathematics  was  organized  into  three  regular 


232  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

classes,  tbe  Juuior,  Intermediate,  and  Senior.  During  part  of  the  time 
it  was  found  necessary  to  estabiisli  also  an  introductory  class  for  stu- 
dents deficient  in  preliminary  studies.  The  requirements  for  admission 
to  the  Junior  class  were  a  knowledge  of  arithmetic-  and  Loomis's  Ele- 
ments of  Algebra.  The  Junior  class  studied  Loomis's  Treatise  on 
Algebra,  and  Loomis's  (later  Wentworth's)  Plane  and  Solid  Geometry. 
The  Intermediate  class  was  taught  Loomis's  Plane  and  Spherical  Trigo- 
nometry, and  Loomis's  Analytical  Geometry.  The  Senior  class  com- 
pleted the  course  in  mathematics  by  the  study  of  Cburch's  Descriptive 
Geometry,  and  Loomis's  Differential  and  Integral  Calculus.  Professor 
Cross  is,  we  believe,  the  first  teacher  who  ever  carried  classes  in  K"ew 
Orleans  through  the  calculus. 

Yery  efficient  work  was  done  by  students  in  the  school  of  physics. 
This  was  in  charge  of  Prof.  Brown  Ayres.  Professor  Ayres  received 
his  general  education  at  the  Washington  and  Lee  University,  and  his 
training  as  a  specialist  at  the  Stevens  Institute  and  the  Johns  Hopkins 
University.  At  the  last  institution  he  was  honored  with  a  fellowship 
in  physics.  He  is  a  true  lover  of  science,  and,  with  great  proficiency 
in  the  theoretical  and  mathematical  parts  of  his  subject,  combines  great 
mechanical  ingenuity  and  skill.  In  his  prelections  on  text-books  he  is 
extremely  clear,  and  his  experiments  are  always  very  successful  and  inter- 
esting. His  great  aim  is  to  awaken  in  students  a  genuine  love  for  pure 
science.  In  his  school  students  had  frequent  opportunities  of  applying 
their  knowledge  of  pure  mathematics  to  physical  problems.  The  theory 
of  the  combination  of  observations  by  the  method  of  least  squares  was 
a  study  in  his  course.  During  several  years  he  taught  also  analytical 
mechanics,  using  the  work  of  De  Yolson  Wood. 

In  1884  the  University  of  Louisiana  was  absorbed  into  the  Tulane 
University  of  Louisiana.  Paul  Tulane,  who  had  been  in  business  in 
l^ew  Orleans  for  fifty  years,  donated  the  greater  part  of  his  large 
fortune  for  higher  education  in  New  Orleans.  Owing  lo  his  munificence, 
Tulane  University  has  the  good  fortune  of  being  free  from  those  pecun- 
iary embarrassments  with  which  the  University  of  Louisiana  had 
always  to  contend.  Under  tbe  presidency  of  Col.  William  Preston 
Johnston,  an  educator  of  great  ability  and  wide  reputation,  the  courses 
of  study  as  they  had  existed  in  the  University  of  Louisiana  were  reor- 
ganized.* Not  trusting  in  the  ability  of  immature  students,  or  even  of 
parents  unaccustomed  to  consider  the  due  proportions  and  sequence  of 
studies,  to*  properly  formulate  their  own  ideals  in  education,  Tulane 
College  offered  a  series  of  six  equivalent  curricula  with  prescribed 
branches,  all  leading  to  the  degree  of  bachelor  of  arts.  These  six  courses 
of  study  were  denominated,  respectivelj^,  the  Classical,  Literary,  Math- 
ematical, Natural  Science,  Commercial,  and  Mechanical  Courses.    In  the 

*  For  ftirtiher  information  regarding  the  plan  and  workings  of  Tulane  University, 
see  President  Johnston's  address  on  "  Education  in  Louisiana/'*  before  tlie  National 
Educational  Convention,  Topeka,  Kan.,  July  15,  ItJ85. 


INFLUX  OF  FEENCH  MATHEMATICS.  233 

spring  of  1880,  the  commercial  course  was  discontinued,  and  the  math- 
ematical course  had  its  name  changed  to  physical  science  course. 

All  the  professors  of  the  University  of  Louisiana  continued  to  hold 
their  respective  chairs  under  the  new  regime.  Several  new  professors 
were  added  to  the  faculty. 

The  mathematical  requirements  for  admission  to  Tulane  College  are 
a  knowledge  of  algebra  to  quadratics  and  of  plane  geometry.  The 
course  in  mathematics  is  the  same  for  all  Freshmen.  After  completing 
the  algebra  they  take  up  solid  geometry,  plane  and  spherical  trigo- 
nometry, surveying  and  leveling,  and  navigation.  In  the  Sophomore 
year,  classical  and  literary  students  pursue  analytical  geometry,  three 
hours  per  week,  before  Christmas.  This  completes  the  mathematics  for 
students  in  those  two  courses.  In  the  three  other  courses  mathematics 
is  pursued  six  hours  per  week  throughout  the  year,  and  consists  in  the 
study  of  analytical  geometry  and  differential  calculus.  In  the  first  half 
of  the  Junior  year,  students  in  the  physical  science  course  and  mechani- 
cal course  pursue  the  study  of  integral  calculus.  These  branches  are 
taught  by  Professor  Cross  from  Loomis's  text-books,  excepting  that 
Wentworth's  book  is  used  in  geometry. 

The  mathematical  teaching  has,  thus  far,  been  strictly  confined  to  the 
ordinary  college  branches.  No  work  of  university  grade,  as  distin- 
guished from  college  grade,  has  yet  been  attempted.  "The  end  kept 
always  in  view  is  to  impress  the  principles  of  mathematical  truth  clearly 
and  deeply  on  the  mind,  by  careful  explanations,  by  daily  examinations, 
and  by  constant  application  of  these  principles  by  the  students  them- 
selves to  numerous  examples  taken  from  the  text- books  and  from  other 
sources."*  Professor  Cross  believes  in  making  a  clear  presentation  to 
the  student  of  the  principles  of  mathematics,  without  applying  them 
to  any  great  number  of  special  cases.  In  his  opinion,  much  valuable 
time  is  wasted  in  the  solution  of  problems.  If  a  student  can  give,  for 
instance,  the  general  solution  of  a  quadratic  equation,  then  there  is  no 
need  of  solving  dozens  of  special  exercises  under  this  head.  In  ge- 
ometry careful  attention  is  given  to  the  correct  understanding  of  the 
demonstrations  given  in  the  book,  but  little  or  no  effort  is  made  to  solve 
original  exercises.  In  the  class  room  Professor  Cross  preserves  strict 
discipline  and  is  earnest  in  the  discharge  of  his  duties.  When  the 
routine  work  of  the  day  is  over,  his  mind  finds  relaxation  and  rest  in  a 
good  game  of  chess  or  checkers. 

Students  in  the  mechanical  and  physical  science  courses  study  an- 
alytical mechanics  under  Professor  Ayres  six  hours  per  week  during 
the  second  half  of  the  Junior  year.  This  subject  has  been  exceedingly 
well  taught.  The  text-book  used  heretofore  in  connection  with  lectures 
has  been  Wood's  Analytical  Mechanics.  This  is  a  good  text-book,  in- 
asmuch as  the  subject  is  taken  up  more  or  less  inductively,  and  a  large 

*  Catalogue  of  the  Tulane  University  of  Louisiana,  1888-89,  p.  46. 


234  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

number  of  special  and  well-graded  problems  is  given  to  be  worked  by 
the  student.  Wood  makes  extensive  use  of  the  calculus  in  his  Analyti- 
cal Mechanics.  The  experience  has  been  at  this  institution,  as  also  at 
others,  that  students  who  have  gone  through  Loomis's  Calculus  are 
hardly  well  enough  prepared  in  that  branch  to  pursue  with  ease  a 
course  in  analytical  mechanics.  Some  important  parts  of  the  integral 
calculus,  particularly  definite  integrals,  receive  exceedingly  meager 
treatment  in  this  book.  The  course  in  analytical  mechanics  serves  to 
impress  more  deeply  and  lastingly  the  principles  of  the  calculus  and 
displays  to  the  student  its  wonderful  power  in  the  solution  of  all  sorts 
of  mechanical  problems.  This  year  (1888-89)  Michie's  Analytical  Me- 
chanics will  be  used  as  a  text-book  by  Professor  Ayres.  It  contains  a 
beautiful  chapter  on  graphical  statics.  In  tlie  Senior  year  students  in 
the  mechanical  course  take  up  the  subject  of  applied  mechanics.  Pro- 
fessor Ayres  is  using,  this  year,  Cotterill's  Applied  Mechanics,  a  stand- 
ard work  of  great  merit. 

In  1883  a  very  fine  collection  of  physical  apparatus  was  purchased 
by  the  university  at  a  great  expense.  In  optics  the  collection  is  excel- 
lent. The  university  is  fortunate  in  having  a  physicist  who  knows 
how  to  make  use  of  delicate  instruments.  Since  the  above  date  Pro- 
fessor Ayres  has  devoted  much  of  his  time  and  energy  toward  building 
up  a  good  laboratory.  A  practical  physical  laboratory  is  somewhat  of 
a  novelty  in  the  South.  Tulane  University  oifers  now  as  good  and  effi- 
cient courses  in  experimental  physics  to  students  of  college  grade  as  any 
university  in  the  country. 

Since  Tulane  University  is  dependent  for  its  supply  of  students 
chiefly  upon  its  own  high  school,  wise  provisions  have  been  made  for 
more  thorough  instruction  in  that  department.  With  Professor  Ashley 
D.  Hurt  as  head-master  the  high  school  has  been  prosperous  and 
thorough  in  its  work.  Both  teachers  and  pupils  are  working  with  great 
earnestness,  and  it  is  gratifying  to  know  that  the  number  of  students 
entering  the  college  after  graduating  from  the  high  school  is  decidedly 
on  the  increase. 

The  New  Orleans  Academy  of  Sciences  holds  its  meetings  at  the 
Tulane  University.  The  professors  of  the  university  are  its  leading 
members.  There  is  a  general  meeting  once  every  month  for  all  mem- 
bers of  the  academy.  In  addition  to  this,  there  are  section  meetings. 
"  Section  A,"  the  mathematical  and  physical  section,  meets  the  second 
Tuesday  of  every  month.  Professor  Ayres  has  been  the  leading  spirit 
in  this  section,  and  has  contributed  many  an  interesting  paper  on 
physics  and  mathematics.  Two  years  ago  the  academy  began  publish- 
ing an  annual  volume,  containing  the  principal  papers  read  during  the 
year.  The  publication  for  the  year  1887-88  contains  an  article  on  the 
'^  History  of  Infinite  Series,"  and  an  interesting  article  by  Professor 
Ayres  on  "  Physics  and  Psychology."    Daring  the  last  two  years  the 


INFLUX  OF  FRENCH  MATHEMATICS.  235 

acadeioy  has  been  iu  a  flonrisliiug  condition,  and  the  quality  of  the 
work  done  has  been  improving  continually. 

In  the  fall  of  1887  the  H.  Sophie  Newcomb  Memorial  College  for 
Young  Women  was  opened  as  a  branch  of  Tulane  University.  It  was 
founded  on  an  endowment  made  by  Mrs.  J.  L.  Newcomb,  of  Xew  York. 
This  institution  is  under  the  able  management  of  President  Brandt  V. 
B.  Dixon,  who  is  also  professor  of  metaphysics  and  mental  science  at 
the  Tulane  University.  It  is  the  aim  to  put  the  Newcomb  College  on 
an  equal  footing  with  the  Tulane  College.  Young  women  will  thus 
have  the  same  facilities  for  higher  education  in  Ne^  Orleans  that 
young  men  have. 

The  first  year  (1887-88)  was  a  year  of  organization.  Many  features 
of  the  school  were  of  necessity  only  tentative.  The  great  obstacle  to 
high  scholarship  is  the  lack  of  proper  preparation  on  the  part  of  appli- 
cants. For  this  reason  it  has  been  necessary  to  establish  a  preparatory 
department.  The  Newcomb  College  offers  four  parallel  and  equivalent 
courses  of  study — the  Classical,  Literary,  Scientific,  and  Industrial.  In 
the  two  preparatory  years,  higher  arithmetic  and  algebra  are  studied 
It  is  the  intention  to  introduce  also  a  course  on  inventional  geometry 
The  first  year  in  college  is  devoted  to  geometry,  the  second  to  the  com 
pletion  of  algebra  and  to  trigonometry.  To  students  taking  the  scien 
tific  and  industrial  courses,  analytical  geometry  is  offered  in  the  Junior 
year,  and  calculus  and  astronomy  in  the  Senior  year.  During  the  first 
year  in  the  history  of  the  college  there  were  classes  in  algebra,  geome- 
try, and  trigonometry.  Wentworth's  test-books  were  used.  In  the  pre- 
paratory department  there  were  two  cla.sses,  one  in  arithmetic  and 
algebra,  and  the  other  in  algebra.  The  latter  class  did  faithful  and 
thorough  work  in  Wentworth's  Complete  Algebra  through  quadratic 
equations.  This  division  did  as  good  work  as  any  class  of  young  men 
which  the  professor  has  taught.  If  not  always  quite  as  penetrating  in 
the  solution  of  problems  as  young  men,  the  young  ladies  worked  more 
faithfully  and  perse veringly.  The  lowest  class  of  college  grade  finished 
plane  geometry  and  then  reviewed  algebra  as  far  as  logarithms.  The 
work  in  geometry  was  quite  satisfactory.  A  great  effort  was  made  to 
induce  students  to  solve  original  exercises.  While  paralogisms  were 
very  frequent,  especially  at  first,  the  efforts  were  not  without  some  suc- 
cess. The  solving  of  original  exercises  in  geometry  is  too  much  neg- 
lected in  our  schools }  nor  are  our  test-books  always  satisfactory  ou  this 
subject.  In  the  opinion  of  the  writer,  the  number  of  exercises  should 
be  greatly  increased,  and  very  great  care  should  be  taken  to  either 
omit  the  difficult  exercises  or  give  "  hints  "  as  to  their  mode  of  solution. 
Students  should  not  be  permitted  to  get  disheartened  in  this  sort  of 
work.  "  The  inventive  power  grows  best  in  the  sunshine  of  encourage- 
ment." Wentworih  has  greatly  improved  his  text-book  in  his  revised 
edition  of  1888,  by  the  insertion  of  seven  hundred  additional  exercises. 


236  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

The  professor  lias  found  that  the  interest  which  pupils  take  in  their 
studies  may  be  increased  if  the  solution  of  problems  and  the  cold  logic 
of  geometrical  demonstrations  are  interspersed  by  historical  remarks 
and  anecdotes.  A  class  in  arithmetic  will  be  pleased  to  hear  about  the 
Hindoos  and  their  invention  of  the  "Arabic  notation;"  they  will  mar- 
vel at  the  thousands  of  years  which  elapsed  before  people  had  even 
thought  of  introducing  into  the  numeral  notation  that  Columbus  egg, 
the  zero ;  they  will  find  it  astounding  that  it  should  have  taken  so  long 
to  invent  a  notation  which  they  themselves  can  now  learnin  a  few  weeks. 
The  class  will  take  an  interest  in  the  history  of  decimal  fractions  and 
the  various  notations  that  were  used  once  in  place  of  our  decimal  point. 
After  the  pupils  have  learned  how  to  bisect  a  given  angle,  surprise  them 
by  telling  of  the  many  futile  attempts  which  have  been  made  to  solve 
by  elementary  geometry  the  apparently  very  simxDle  problem  of  the  tri- 
sectiou  of  an  angle.  When  they  know  how  to  construct  a  square  whose 
area  is  double  the  area  of  a  given  square,  tell  them  about  the  duplica- 
tion of  the  cube — how  the  wrath  of  Apollo  could  be  appeased  only  by 
the  construction  of  a  cubical  altar  double  the  given  altar,  and  how 
mathematicians  long  wrestled  with  this  problem.  After  the  class  have 
exhausted  their  energies  on  the  theorem  of  the  right-angled  triangle,  tell 
them  something  about  its  discoverer — how  Pythagoras,  jubilant  over 
his  great  accomplishment,  [is  said  to  have]  sacrificed  a  hecatomb  to  the 
Muses  who  inspired  him.  When  the  value  of  mathematical  training 
is  called  in  question,  quote  the  inscription  over  the  entrance  into  the 
academy  of  Plato,  the  philosopher:  "Let  no  one  who  is  unacquainted 
with  geometry  enter  here."  To  more  advanced  students  the  history  of 
mathematics  becomes  instructive  and  profitable  as  well  as  interesting. 
It  seems  to  me  that  students  in  analytical  geometry  should  know  some- 
thing of  Descartes,  who  originated  this  branch  of  geometry,  that,  tak- 
ing up  differential  and  integral  calculus,  they  should  become  familiar 
with  the  parts  which  E"ewton,  Leibnitz,  and  Lagrange  played  in  creat- 
ing the  transcendental  analysis.  No  one  can  claim  to  have  a  fair  knowl- 
edge of  this  subject  who  knows  not  something  about  the  three  methods 
taught  by  these  great  analysts.  In  his  historical  talk  it  is  possible  for 
the  teacher  to  make  it  plain  to  the  student  that  mathematics  is  not  a 
dead  science  in  which  no  new  discoveries  are  or  can  be  made,  but  that 
it  is  a  living  science  in  which  racing  progress  is  being  made  all  the 
time. 

UNIVEESITY  OP   TEXAS. 

The  University  of  Texas  opened  its  doors  to  students  for  the  first 
time  in  1882.  The  first  professor  of  mathematics  was  Leroy  Brown, 
who  served  one  year.  He  was  succeeded  by  G.  B.  Halsted  as  professor 
of  pure  and  applied  mathematics.  At  the  same  time  with  Halsted,  A. 
V.  Lane  was  elected  assistant  instructor  in  mathematics.  He  was  ad- 
vanced to  the  position  of  assistant  professor  of  applied  mathematics 
in  1885. 


INFLUX  OF  FRENCH  MATHEMATICS.  237 

Prof.  G.  B.  Halsted  was  graduated  in  Princeton  in  1875,  and  received 
the  degree  of  doctor  of  philosophy  at  the  Johns  Hopkins  University  in 
1879,  where  he  had  studied  for  two  years  under  Professor  Sylvester,  and 
had  held  a  fellowship  in  mathematics.  Before  taking  his  degree  he 
spent  some  time  in  Berlin,  prosecuting  mathematical  studies.  In  1878 
he  was  appointed  tutor  in  mathematics  at  Princeton  College,  and  three 
years  later  instructor  in  post-graduate  mathematics. 

Dr.  Halsted  has  established  a  wide  reputation  as  a  mathematician 
and  logician.  He  has  contributed  to  the  American  Journal  of  Mathe- 
matics, the  Annals  of  Mathematics,  the  Mathematical  Magazine,  the 
Euglish.  Philosophical  Magazine,  and  several  other  scientific  journals. 
He  has  published  two  books,  An  Elementary  Treatise  on  Mensuration 
(Boston,  1881),  and  The  Elements  of  Geometry  (N"ew  York,  1885).  His 
books  and  scientific  articles  have  been  favorably  reviewed  in  leading 
foreign  journals.  His  Metrical  Geometry  (mensuration)  is  the  best  book 
of  its  kind  that  has  been  published  in  this  country.  It  contains  many 
new  and  interesting  features.  Of  these  we  would  mention  his  treat- 
ment of  solid  angles  (the  words  steregon  and  steradian^  now  quite  gen- 
erally adopted,  were  manufactured  by  him  and  first  used  here)  and  his 
discussion  of  the  prismatoid,  deriving  a  general  formula  for  its  volume. 
He  introduced  a  distinction  between  the  words  spliere  and  glohe  (mak- 
ing one  to  mean  a  surface  and  the  other  a  solid),  which  is  worthy  of 
general  adoption. 

The  distinguishing  feature  of  the  two  works  of  Halsted  is  their  sci- 
entific rigor.  Teachers  who  favor  a  rigid  treatment  of  geometry  will 
find  it  in  his  Elements.  The  book  rejects  the  "  directional  method  "  as 
wholly  unscientific ',  also  the  use  of  the  word  "  distance  "  as  a  funda- 
mental geometric  concept.  The  word  sect,  first  used  in  his  Mensuration, 
is  introduced  here,  meaning  "  the  part  of  a  line  between  two  definite 
points."  Many  teachers  do  not  endorse  the  introduction  of  this  new 
technical  term  in  elementary  geometry,  as  they  think  that  there  is  no 
particular  call  for  it.  The  author  is  certainly  right  in  protesting  against 
the  use  of  the  word  "  distance"  in  two  different  senses.  That  there  has 
really  been  a  want  for  some  of  the  other  new  technical  terms  first  in- 
troduced by  Halsted  is  evident  by  the  fact  that  they  have  been  adopted 
in  standard  works,  such  as  the  Encyclopaedia  Britannica. 

Like  his  Mensuration,  his  Elements  of  Geometry  possesses  many 
novelties.  In  his  book  on  Rectangles  he  introduces  a  strictly  geometric 
algebra,  where  a  and  h  mean  sects,  and,  by  definition,  ah  means  their 
rectangle,  thus  avoiding  measurement  and  the  use  of  numbers.  Eatio 
and  proportion  are  strictly  treated,  but  without  limits.  The  book  on 
two-dimensional  spherics  gives  a  novel  method  of  treating  spherics. 
His  demonstration  of  the  two-term  prismoidal  formula  has  been  trans- 
lated into  French  by  the  editors  of  a  mathematical  journal  published  in 
Belgium.  Halsted  is  the  first  writer  in  this  country  to  preface  a  geom- 
etry by  a  preliminary  chapter  on  logic.    Judged  from  a  scientific  jDoint 


238  TEACriING   AND    HISTORY    OF   MATHEMATICS. 

of  view,  we  believe  Halsted's  Geometry  to  be  the  peer  of  any  geometry 
published  iu  America. 

Professor  Lane  has  contributed  one  article  on  ''Eoulettes"  to  the 
American  Journal  of  Mathematics,  and  has  written  a  neat  little  book 
on  Adjustments  of  the  Compass,  Transit  and  Level.  Professor  Lane 
taught  chiefly  the  applied  mathematics,  i.  e.,  mathematics  applied  to 
engineering,  and  reached  good  results  in  his  work.  In  June,  1888,  he  re- 
signed his  professorship,  and  his  place  was  filled  by  the  selection  of  a 
native  Texan,  T.  U.  Taylor.  Professor  Taylor  is  a  graduate  of  the  Uni- 
versity of  Virginia,  and  before  accepting  the  present  position  was  pro- 
fessor of  pure  and  applied  mathematics  in  the  Miller  Manual  Labor 
School  of  Virginia. 

The  mathematical  requirements  for  admission  have  been  from  the 
beginning  the  same  as  they  are  now,  except  that  Prof.  L.  Brown  ex- 
amined students  in  Weutworth's  Geometry  instead  of  Halsted's.  As 
stated  in  the  catalogue  of  1887-88,  the  terms  for  admission  are  as  fol- 
lows :  "Arithmetic,  including  proportion,  decimals,  interest,  discount, 
and  the  metric  system ;  algebra,  including  theory  of  exponents,  radicals, 
simple  and  quadratic  equations ;  and  the  elements  of  plane  geometry 
(corresponding  to  the  first  six  books  of  Halsted's  Geometry). 

"Passing  these  examinations,  a  student  will  be  admitted  to  the 
Freshman  class  in  the  course  of  science,  or  to  the  Junior  cl?iss  of  the 
law  department." 

Great  efforts  are  being  made  to  cause  the  high  schools  in  the  State  to 
work  in  line  with  the  university.  High  schools  desiring  the  privilege 
of  sending  their  graduates  to  the  university  without  examination  are 
inspected  by  committees  from  the  faculty  of  the  university,  and  if  the 
work  of  a  school  be  found  satisfactory  the  school  is  "  approved."  Thus 
far  the  number  of  irregular  students  in  the  academical  department  of 
the  university  has  been  large,  but  as  the  institution  grows  older,  the 
students  entering  with  a  view  of  taking  a  four-years'  course  and  grad- 
uating will  doubtless  rapidly  increase. 

During  the  first  year  of  the  university  there  were,  naturally,  no 
claw^ses  formed  in  the  higher  mathematics.  At  the  beginning  of  the 
second  year,  in  addition  to  the  lower  classes,  there  was  a  Sophomore 
class  iu  analytic  geometry,  and  a  Junior  class  in  differential  and  inte- 
^gral  calculus.  At  the  beginning  of  the  third  year,  in  addition  to  these, 
there  was  a  Senior  class  in  quaternions,  and  since  then  there  have 
always  beeu  Freshman,  Sophomore,  Junior,  and  Senior  classes  in  math- 
ematics. 

At  the  beginning,  Wentworth's  Algebra  and  Geometry  were  used  by 
Professor  Brown.  When  Professor  Halsted  entered  upon  his  duties 
at  this  university  he  "found  that  the  lack  of  rigor  iu  Wentworth's  Geom- 
etry was  so  exasperating"  that  he  "  could  not  continue  to  use  it  with 
comfort  or  a  clear  conscience,"  and  so  he  put  in  form  for  the  printer  his 
Dwu  manuscript  on  geometr5%    His  geometry  has  been  used  since  its 


INFLUX  OF  FRENCH  MATHEMATICS.  239 

issue,  supplemented  by  Halsted's  Mensuration.    The  analytic  geometry 
used  is  Puckle's  Conic  Sections.    Until  the  present  year  (1888-89)  By- 
erly's  Calculus  lias  been  taugiit.    Post-graduate  courses  in  mathematics 
are  now  offered  to  students. 
The  present  mathematical  course  is  as  follows  (catalogue  1887-88) : 

The  Freshman  class  will  study  algebra,  solid  geometry,  spherics,  mensuration,  plane 
and  spherical  trigonometry,  with  their  applications  to  surveying,  navigation,  etc. 

The  Sophomore  class  will  study  analytical  geometry,  graphic  algebra,  and  theory  of 
equations. 

The  Junior  class  will  study  analytical  geometry  of  three  dimensions,  differential 
and  integral  calculus.  This  course  of  study  will  embrace  the  applications  of  the  cal- 
culus to  mechanics  and  physics. 

The  Senior  class  will  study  determinants,  quaternions,  invariants,  and  quan- 
tics,     *     *     * 

In  the  higher  classes  will  be  discussed  the  history  and  logical  structure  of  the  math- 
ematical sciences,  and  the  logical  theory  of  the  calculus,  the  theory  of  limits,  and  the 
infinitesimal  method. 

Text-ioolcs. — Wentworth's  Complete  Algebra ;  Halsted's  Geometry  (John  Wiley  & 
Sons,  New  York) ;  Halsted's  Mensuration,  3d  Ed.  (Giun  &  Co.) ;  Wentworth's  Trig- 
onometry, Surveying,  and  Navigation;  Graphic  Algebra,  by  Phillips  «fc  Beebe; 
Puckle's  Conic  Sections,  5th  Ed. ;  Smith's  Solid  Geometry ;  Newcomb's  Differential 
and  Integral  Calculus ;  Theory  of  Equations,  by  Burnside  and  Panton,  2d  Ed. ; 
Muir's  Determinants ;  Scott's  Determinants ;  Salmon's  Modern  Higher  Algebra,  4th 
Ed.  ;  Hardy's  Quaternions. 

Engineering  students  are  required  to  take  the  four-years'  course ; 
science  students,  the  studies  for  the  first  three  years ;  arts  students, 
those  of  the  first  two  years ;  and  letters  students,  those  of  the  first  year. 

Two  post-graduate  courses  are  offered  : 

I.  A  course  preparatory  to  original  investigation  in  the  objective  sciences.  This 
will  include  infinitesimal  calculus,  the  method  of  least  squares,  kinematic,  liukage, 
difierential  equations,  the  calculus  of  finite  differences. 

Text  tools. — Williamson's  Differential  Calculus,  Williamson's  Integral  Calculus, 
Clifford's  Kinematic,  Forsyth's  Differential  Equations,  Boole's  Differential  Equations, 
Boole's  Calculus  of  Finite  Differences,  Merriman's  Method  of  Least  Squares. 

II.  A  course  preparatory  to  original  investigation  in  the  subjective  sciences.  This 
will  include  projective  geometry,  the  theory  of  numbers,  the  algebra  of  logic,  the 
theory  of  probability,  non-Euclidian  geometry. 

Text-loolcs. — Cremona's  Projective  Geometry;  Lejeune  Dirichlet's  Zahlentheorie,  3d 
Ed.;  Macfarlane's  Algebra  of  Logic ;  Boole's  Laws  of  Thought;  Todhunter's  History 
of  the  Theory  of  Probability  ;  Frischauf's  Absolute  Geometric. 

The  catalogue  for  1887-88  gives  one  student  taking  post-graduate 
studies  in  mathematics. 

The  university  is  open  to  both  sexes.  "A  number  of  young  ladies 
still  show  that  they  are  capable  of  mastering  even  the  abstruse  modern 
developments  of  this  oldest  of  the  sciences."  (Professor  Halsted,  June, 
1888.) 

WASHINGTON  UNIVEESITT. 

Up  to  the  date  of  writing  we  have  not  been  able  to  secure  the  infor- 
mation desirable  for  a  sketch  of  the  mathematical  teaching  at  this  uni- 
versity, but  an  excellent  biographical  notice  of  Professor  William 


2'40  TEACHING   AND    HISTORY   OF   MATHEMATICS. 

Chauvenet,  the  first  professor  of  mathematics  at  Washington  Univer- 
sity, has  been  written  for  us  by  his  son,  Eegis  Chauvenet,  now  presi- 
dent of  the  State  School  of  Mines,  at  Golden,  Colo.  Professor  William 
Chauvenet  ranks  among  the  coryphaei  of  science  in  America.  He  and 
Benjamin  Peirce  have  done  more  for  the  advancement  of  mathematical 
and  astronomical  science,  and  for  the  raising  to  a  higher  level  of  the  in- 
struction in  these  subjects,  than  any  other  two  Americans.  It  is  our 
wish,  on  that  account,  to  place  before  the  reader  a  somewhat  full  sketch 
of  the  life  and  works  of  Professor  William  Chauvenet.  The  biograph- 
ical notice  above  referred  to  is  as  follows : 

"  William  Marc  Chauvenet,  father  of  the  subject  of  this  sketch,  was 
born  at  Narbonne,  France,  in  1790,  and  came  to  the  United  States  in 
1816.  He  was  the  youngest  of  four  brothers,,  another  of  whom  also 
came  to  this  country  but  has  left  no  descendants.  William  Marc  was 
a  man  of  education  and  culture,  versed  in  several  languages,  and  a  con- 
stant reader.  He  came  to  America,  however,  in  connection  with  a  manu- 
facturing enterprise  which  had  its  headquarters  in  New  York,  with  a 
branch  at  Boston.  The  latter  department  was  under  Mr.  Chauvenet's 
charge,  and  here  he  married,  in  1819,  Miss  Mary  B.  Kerr,  of  Eosbury. 
This  was  before  a  heavy  defalcation  in  the  Kew  York  house,  which 
broke  up  the  enterprise  so  badly  that  all  investments  in  it  proved  to  be 
total  losses.  Mr.  Chauvenet  having  an  idea  that  rural  life  would  suit 
his  taste,  bought  a  small  farm  close  to  Milford,  Pike  County,  Pa.,  and 
it  was  here  that  his  only  child,  William  Chauvenet,  was  born,  May  24, 
1820. 

"  By  the  advice  of  friends  Mr.  Chauvenet  soon  gave  up  his  attempt 
at  farming,  and  settled  in  Philadelphia,  where  his  son  grew  to  man- 
hood. His  rapid  progress  at  school  attracted  such  attention  from  his 
instructors,  especially  in  mathematics,  that  his  father  easily  yielded  to 
their  advice,  and  sent  him  to  Yale  College,  where  he  graduated  in  1840, 
^facile  princeps '  in  mathematics,  and  high  in  standing  in  all  other 
branches.  The  honorary  societies,  '  Phi  Delta  Kappa '  and  '  Chi  Delta 
Theta,'  denoting  respectively  the  fifteen  of  highest  standing  and  the  fif. 
teen  best  writers  of  the  class,  each  claimed  him  as  a  member. 

"  Upon  his  return  to  his  home  he  was,  after  a  brief  incumbency  in  a 
subordinate  positioo,  appointed  professor  of  mathematics  in  the  Navy. 
Late  in  1841  ho  married  Miss  Catherine .  Hemple,  of  Philadelphia. 
Shortly  after  this  ho  served  a  brief  term  on  a  United  States  vessel,  as 
instructor  to  midshipmen,  but  did  not  go  upon  a  foreign  cruise,  and  was 
soon  detailed  to  the  'Naval  Asylum,'  then  situated  at  Philadelphia. 
Here  midsbi])men  were  sent  at  that  time,  to  receive  instruction  and 
examinations,  principally  in  mathematics  and  the  theory  of  navigation. 
The  young  professor  was  struck  with  the  imperfections  in  the  education 
of  naval  officers,  and  it  was  very  largely  through  his  efforts,  aided  by 
such  influences  as  he  could  bring  to  bear  on  the  matter,  that  a  commis- 
sion was  appointed  to  draft  a  plan  for  a  fixed  '  Naval  Academy,'  corre- 


INFLUX  OP  FRENCH  MATHEMATICS.  241 

spending  to  the  Military  Academy  at  West  Point.  Six  naval  officers 
constituted  this  commission,  Professor  Chauvenet  being  of  the  number. 
The  appointment  of  so  young  a  man  (he  was  but  twenty-four  at  the 
time)  on  a  commission  of  such  importance  indicates  what  must  have 
been  his  record,  and  the  impression  he  made  upon  his  seniors  in  years 
and  rank. 

"  The  Naval  Academy  was  formally  called  into  existence  in  the  year 
1845,  being  located  at  Annapolis,  Md.  Professor  Chauvenet  was  ap- 
pointed to  the  chair  of  mathematics,  and  resided  at  the  academy  until 
his  resignation  from  the  Navy  in  1859. 

"  It  was  not  long  after  this  change  of  residence  that  he  began  to  plan 
his  work  on  trigonometry,  which  was  published  in  1850.  Its  title,  'A 
Treatise  on  Plane  and  Spherical  Trigonometry,'  partly  indicated  that 
it  was  not  a  students'  class-book  merely,  but  that  it  took  up  most  of  the 
more  advanced  applications  of  the  subject.  It  soon  assumed  the  posi- 
tion it  still  retains  as  the  standard  reference  work  in  its  line. 

"  Some  time  before  this  publication,  Professor  Chauvenet  had  per- 
suaded his  father  to  retire  from  business  and  accept  a  position  at  the 
.-icademy.  He  came  as  instructor  in  the  French  language,  and  remained 
at  his  post  until  his  death  in  1855. 

''  It  having  been  decided  to  erect  an  astronomical  observatory  at  the 
academy,  Professor  Chauvenet  was  made  professor  of  astronomy  and 
put  in  charge  of  the  observatory.  As  he  became  more  and  more  inter- 
ested in  his  work,  the  idea  of  his  next  treatise,  'Spherical  and  Practical 
Astronomy,'  grew  upon  him,  and,  just  previous  to  his  resignation,  had 
assumed  such  form  that  he  issued  a  prospectus  for  its  publication  as  a 
subscription  work.    This  was  never  carried  out. 

"  In  1859  he  was  notified  that  his  application  for  the  professorship  of 
mathematics  at  Yale  College  would  be  followed  by  his  election  to  that 
position.  Almost  simultaneously  with  this  came  a  call  to  St.  Louis, 
Mo.,  where  he  was  offered  the  same  chair  in  the  then  newly-established 
Washington  University.  After  much  deliberation  he  accepted  the 
latter,  and  removed  with  his  family  (including  at  that  time  his  mother) 
to  St.  Louis,  in  the  fall  of  1859. 

"^  Chancellor  Hoyt,  who  was  at  the  head  of  the  '  Washington '  at  this 
time,  died  early  in  the  '  sixties,'  and  Professor  Chauvenet  was  elected 
to  the  vacancy.  He  still  continued  his  duties  as  professor  of  mathematics, 
however,  and  now  resumed  his  work  on  the  'Astronomy.'  The  risks 
of  publication  were  great,  and  his  means  did  not  enable  him  to  guar- 
antee the  publishers  agaiusfc  loss.  The  Civil  War  was  in  progress,  and 
the  time  seemed  inopportune  for  such  an  undertaking.  It  was  to  the 
liberality  of  certain  friends,  chiefly  to  the  initiative  of  Mr.  (afterward 
Judge)  Thomas  T.  Gantt,  of  the  St.  Louis  bar,  that  a  guarantee  fund 
was  raised,  sufficient  in  the  opinion  of  the  publishers  to  prevent  any 
loss  to  them.    The  work,  in  two  octavo  volumes,  was  published  in  1863. 

"  Few  works  of  a  scientific  nature,  by  American  authors,  have  been 
881— No.  3 16 


242  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

received  with  snch  universal  favor,  by  those  competent  to  judge  of  its 
merits,  as  was  this.  Its  reputation  was  quite  as  great  in  Europe  as 
here,  while  of  course  it  is  not  (as  it  was  never  intended  to  be)  a  treatise 
much  known  outside  of  scientific,  and  more  especially  astronomical,  cir- 
cles. Its  scope,  and  the  rigorous  methods  adopted,  are  sufficiently 
indicated  in  the  author's  preface.  It  retains  to-day  its  standard  char- 
acter, as  fully  as  when  this  was  first  recognized  by  the  scientific  world 
upon  its  publication. 

"Professor  Chauvenet's  mother  died  in  St.  Louis,  not  long  after  the 
appearance  of  the  Astronomy,  and  it  was  but  a  few  months  later  that 
the  first  symptoms  of  the  disease  that  proved  finally  fatal  to  him,  made 
their  appearance.  Partial  recovery  and  resumption  of  his  duties  was 
followed  by  a  long  period  of  alternating  hopes  and  fears,  during  which 
time  he  tried  in  vain  different  parts  of  the  United  States,  from  South 
Carolina  to  Minnesota.  During  this  illness  he  worked  at  his  only  ele- 
mentary publication,  the '  Geometry,'  which  he  undertook,  partly  because 
he  had  long  thought  that  the  popular  texts  of  the  day  were  marked  by 
too  strict  an  adherence  to  strictly  '  Euclidian '  methods,  and  partly  be- 
cause he  wished  to  provide  an  income  for  his  family,  by  the  publication 
of  a  text  for  which  he  had  reason  to  suppose  there  would  be  a  larger 
sale  than  was  possible  with  advanced  treatises.  The  publication  of 
this  work  shortly  antedated  his  death,  which  occurred  at  St,  Paul, 
Minn.,  December  13,  1870. 

"  Professor  Chauvenet  left,  so  to  speak,  two  distinct  impressions  be- 
hind him.    By  far  the  larger  circle,  in  numbers,  of  those  who  knew  him, 
were  of  those  to  whom  his  scientific  attainments,  though  known,  were  as 
traditions  merely,  since  they  were  in  a  field  whose  extent  was  to  them 
only  a  matter  of  vague  conjecture.    To  these  he  left  the  impression  of  a 
man  of  wide  and  varied  culture,  and  keen  critical  taste.    Probably  few 
scientists  of  distinction  were  more  keenly  interested  in  lines  outside  of 
their  own  specialties.    He  was  not  only  a  critic  in  music,  but  to  his 
latest  day  a  pianist  of  no  mean  ability,  always  expressing  a  preference, 
in  his  own  playing,  for  the  works  of  Beethoven,  which  he  rendered  with 
an  interpretation  which  never  failed  to  excite  the  admiration  of  musi- 
cians whose  execution  surpassed  his  own.    His  knowledge  of  English 
literature  was  extensive,  but  he  read  and  re-read  a  few  authors,  at  least 
in  the  latter  part  of  his  life,  and  his  great  familiarity  with  many  of  these 
gave  point  to  the  old  adage,  'fear  the  man  of  few  books,'  though  perhaps 
not  in  the  sense  in  which  these  words  were  originally  intended.     He  was 
a  ready  writer,  and  contributed  at  times  reviews,  partly  scientific,  to 
various  journals.    His  style  was  clear  and  unaffected,  while,  in  the  re- 
view of  a  pretentious  or  ignorant  author,  he  had  the  gift  of  a  delicate 
sarcasm,  so  light  at  times  as  only  to  be  visible  to  one  reading  between 
the  lines.    For  other  pretenders  he  could  drop  this  mask,  and  write  with 
severity ;  but  only  twice  in  his  life,  to  the  knowledge  of  the  present 
writer,  did  he  ever  do  so.    In  addition  to  his  more  important  writings, 


INFLUX  OF  FRENCH  MATHEMATICS.  243 

he  was  the  author  of  a  '  Lunar  Method,'  still  used  in  the  iN'avy,  and  in- 
vented a  device  called  the '  great  circle  protractor/  by  which  the  naviga- 
tor is  enabled  (knowing  his  position)  to  lay  down  his  course  on  a  'great 
circle'  of  the  globe,  without  further  calculation.  Tiiis  invention  was 
purchased  by  the  United  States  Government  not  long  after  the  close  of 
the  Civil  War. 

"  Professof  Chauvenet's  scientific  reputation  needs  little  comment  on 
the  part  of  the  present  writer.  He  was  one  of  a  group  of  scientists  in 
his  own  or  cognate  lines,  who  were  the  first  to  secure  recognition  abroad, 
as  well  as  at  home,  for  the  position  of  the  exact  sciences  in  the  United 
States.  Among  his  more  intimate  scientific  friends  were  Benjamin 
Peirce  and  Wolcott  Gibbs  (Harvard),  Dr.  B.  A.  Gould,  and  many  others 
whose  names  are  as  household  words  in  the  history  of  scientific  prog- 
ress in  this  country.  At  the  formation  of  the  National  Academy  of 
Sciences  he  was  one  of  the  prominent  members.  But  while  his  scientific 
reputation  will  outlast  his  personal  memory,  it  is  doubtful  if  to  those 
who  knew  him,  even  of  his  scientific  associates,  it  will  ever  be  as  pres- 
ent as  his  strong  personal  attractiveness,  the  result  at  once  of  an  easy 
and  varied  culture,  and  of  a  simple  dignity  of  character,  which  im- 
pressed alike  his  family,  his  friends,  and  his  pupils.  His  family,  con- 
sisting at  the  time  of  his  death  of  his  wife,  four  sons,  and  a  daughter, 
are  all  still  living  (1889)." 

The  only  mathematical  book  written  by  Chauvenet  and  not  mentioned 
in  the  above  sketch  is  a  little  book  entitled  Binomial  Theorem  and 
Logarithms,  published  in  1843  for  the  use  of  midshipmen  at  the  ll^aval 
School,  Philadelphia. 

As  regards  the  quality  of  Professor  Chauvenet's  books,  Prof.  T.  H. 
Safford,  of  Williams  College,  says :  "  This  excellent  man  and  lucid 
writer  was  admirably  adapted  to  promote  mathematical  study  in  this 
country.  His  father,  a  Frenchman  of  much  culture,  trained  him  very 
thoroughly  in  the  knowledge  of  the  French  language,  even  in  its  niceties. 
They  habitually  corresponded  in  that  language ;  and  the  son  was  en- 
abled to  study  the  mathematical  writings  of  his  ancestral  country  in  a 
way  which  enabled  him  to  reproduce  in  English  their  ease  and  grace  of 
style,  as  well  as  their  matter.  In  these  respects  his  works  are  far  more 
attractive  than  those  of  ordinary  English  writers  j  his  Trigonometry  is 
much  the  best  work  on  the  subject  which  I  know  of  in  any  language ; 
his  Spherical  and  Practical  Astronomy  is  frequently  quoted  by  eminent 
continental  astronomers ;  and  his  Geometry  has  raised  the  standard  of 
our  ordinary  text-books,  of  which  it  is  by  far  the  best  existing."* 

Chauvenet's  books,  especially  his  Geometry,  have  been  used  in  the 
best  of  our  schools.  Eecently  a  revised  edition  of  his  Geometry  has  been 
brought  out  by  Professor  Byerly,  of  Harvard.  Among  the  chief  modi- 
fications made  by  him  are  the  following:  (1)  The  " exercises,"  which 

*  Mathematical  Teaching,  by  Prof.  T.  H.  Safford,  1887,  p.  9, 


244  TEACHING   AND    HISTOEY    OP   MATHEMATICS. 

in  tbe  original  are  at  the  end  of  the  book,  are  most  of  them  placed  in 
direct  connection  with  the  theorems  which  they  serve  to  illustrate.  (2) 
The  admirable  little  chapter  in  the  original  edition  on  *'  Modern  Geom- 
etry "  is  omitted.  (3)  The  ''directional  method"  is  introduced.  The 
first  is,  no  doubt,  a  change  for  the  better  j  the  second  and  third  are,  we 
think,  to  be  regretted.  It  seems  to  us  that  the  day  has  come  when  a 
college  course  should  set  aside  some  little  time  to  the  study  of  modern 
methods  in  elementary  geometry,  and  not  confine  itself  to  the  ancient. 
The  introduction  of  the  "  directional  method,"  in  our  opinion,  robs  the 
book  of  some  of  that  admirable  rigor  for  which  the  original  work  of 
Chauvenet  is  so  justly  celebrated. 

His  Trigonometry  and  Astronomy  are  the  first  American  works  to 
introduce  the  consideration  of  the  general  spherical  triangle,  in  which  the 
six  parts  of  the  triangle  are  not  subjected  to  the  condition  that  they 
shall  each  be  less  than  180°,  but  may  have  any  values  less  than  360°. 
This  feature  is  mainly  due  to  Gauss.  The  methods  of  investigation  fol- 
lowed in  these  two  books  are  chiefly  those  of  the  German  school,  of 
which  Bessel  was  the  head. 

UNIVEESITT  OF  MICHIGAN.* 

The  University  of  Michigan  opened  in  1841 .  In  its  organization  Prus- 
sian ideas  predominated.  But  the  regime  which  existed  during  the  first 
ten  years  in  the  history  of  the  university  did  not  prove  efficient.  A 
re-organization  was  therefore  effected  in  1852.  The  board  of  regents 
were,  from  that  time  on,  rendered  independent  of  the  Legislature  by 
intrusting  their  election  to  the  people.  The  German  method  of  govern- 
ing the  faculty  by  an  annual  president  elected  by  that  body  was  aban- 
doned in  1852,  and  it  was  henceforth  the  duty  of  the  board  of  regents 
to  appoint  a  chancellor  for  the  university. 

The  first  appointment  to  a  professorship  attheUniversity  of  Michigan 
was  that  of  George  Palmer  Williams,  in  1841.  He  was  first  assigned  to 
the  chair  of  ancient  languages.  On  the  work  of  this  department,  how- 
ever, he  did  not  enter,  but  exchanged  it  for  that  of  mathematics  and 
natural  philosophy. 

Professor  Williams  was  born  in  Woodstock,  Vt.,  in  1802.  After  grad- 
uating at  the  University  of  Vermont  he  studied  theology  at  Andover, 
then  became  tutor  at  Kenyon  College,  and  later  professor  of  languages 
in  the  Western  University  of  Pennsylvania.  Thence  he  returned  to 
Kenyon  College,  where  he  remained  until  1837,  when  he  entered  upon 
the  services  of  the  board  of  regents  of  the  University  of  Michigan,  as 
principal  of  the  Pontiac  Branch. 

At  the  University  of  Michigan  he  was  professor  of  mathematics  and 
natural  philosophy  until  1854,  professor  of  mathematics  from  1854  to 

*  For  part  of  the  inforraatiou  herein  contained  we  are  indebted  to  Prof.  W.  W. 
Bemau,  of  Ann  Arbor.  The  writer  is  also  under  obligation  to  Charles  E.  Lowreyi 
ph.  D.,  for  interesting  oral  communications. 


INFLUX  OF  FRENCH  MATHEMATICS.  245 

1863,  and  professor  of  physics  from  1863  to  1875.  Williams  was  a  man 
)f  culture  and  refinement,  and  understood  well  the  branches  which  he  at- 
tempted to  teach.  As  an  instructor  he  lacked  thoroughness.  ' '  Though 
'.\iQ  never  felt  himself  called  upon  to  force  the  reluctant  mind  into  a 
thorough  understanding  of  that  for  which  it  had  no  liking,  he  helped 
those  who  desired  to  study  in  attaining  to  the  established  standard, 
and,  in  a  private  way,  he  loved  to  aid  those  who  desired  his  help  in 
branscendingthat  limit.  Astronomy,  though  not  nominally  in  his  profes- 
sorship, he  taught  until  the  revision  of  the  course  in  1854,  and  a  great 
Bnthusiasm  was  annually  awakened  among  the  students  as  they  came 
to  the  calculation  of  eclipses."* 

The  mathematical  requirements  for  admission  were,  in  1847,  arith- 
metic, and  algebra  through  simple  equations.  The  college  course  for 
that  year  included  algebra,  geometry,  conic  sections,  plane  and  spheri- 
eal  geometry,  and  calculus.  In  1848  it  was  the  same,  save  calculus  or 
analytical  geometry,  and  in  1849  calculus  and  analytical  geometry.  The 
text-books  were  those  of  Professor  Davies,  of  West  Point. 

Before  its  reorganization,  in  1852,  "the  institution  had  flagged  some- 
what in  popular  interest ;  the  number  of  its  students  had  fallen  ofl' ;  a 
more  vigorous  and  aggressive  leadership  was  imperatively  needed."  t 
In  the  year  just  named,  Dr.  Henry  P.  Tappau,  of  New  York,  was  inau- 
gurated first  chancellor.  His  connection  with  the  university  marks  a 
new  era  in  its  history.  During  the  reconstruction,  German  ideals  were 
constantly  kept  in  view.  He  thoroughly  understood-  the  workings  of 
German  universities  and  was  a  recognized  champion  among  us  of  uni- 
versity education,  as  distinguished  from  college  education.  In  the  first 
catalogue  (1852-53)  issued  by  him,  we  read :  "An  institution  can  not 
deserve  the  name  of  a  university  which  does  not  aim,  in  all  the  ma- 
terial of  learning,  in  the  professorships  which  it  establishes,  and  in  the 
whole  scope  of  its  provisions,  to  make  it  possible  for  every  student  to 
study  what  he  pleases  and  to  any  extent  he  pleases.  It  is  proposed, 
therefore,  at  as  early  a  day  as  practicable,  to  open  courses  of  lectures 
for  those  who  have  graduated  at  this  or  other  institutions,  and  for 
those  who  in  other  ways  have  made  such  preparation  as  may  enable 
them  to  attend  upon  them  with  advantage.  These  lectures,  in  accord- 
ance with  the  educational  systems  of  Germany  and  France,  will  form 
the  proper  development  of  the  university,  in  distinction  to  the  college 
or  gymnasium  now  in  operation."  The  university  system  has  been 
growing  at  Ann  Arbor,  though  at  first  very  slowly. 

The  first  fruits  of  the  plan  laid  down  in  the  catalogue  just  named  was 
the  appointment  to  the  chair  of  astronomy,  in  1854,  of  Dr.  Francis 
Briinnow,  of  Leipsic,  a  favorite  pupil  and  assistant  of  the  celebrated 
astronomer  Bncke.  Briinnow  remained  at  the  university  until  1863, 
when  he  resigned  to  take  charge  of  the  Dudley  Observatory.    Later,  he 

'University  of  Michigan,  by  Andrew  T,  Brook,  1875,  p.  298. 

t  The  Study  of  History  in  American  Colleges,  by  Herbert  B.  Adams,  p.  90.   ' 


246  TEACHING   AND    HISTORY    OF   MATHEMATICS, 

became  director  of  the  Eoyal  Observatory  in  Dablin,  Ireland.  Under 
his  able  management  the  observatory  at  the  University  of  Michigan 
(called  the  Detroit  Observatory,  in  recognition  of  the  liberality  of  citi- 
zens of  Detroit  who  founded  it)  soon  rose  to  high  rank.  Besides  the 
"  Tables  of  Flora"  and  the  "  Tables  of  Victoria,"  published  at  Ann 
Arbor,  Dr.  Briinnow  contributed  to  science  his  large  work  on  Spherical 
Astronomy  and  many  papers  on  astronomical  subjects.  But  the  influ- 
ence of  its  renowned  scholar  was  felt  also  in  the  department  of  pure 
mathematics.  It  is  he  who  gave  the  university  its  start  mathematic- 
ally. When  Professor  Olney  became  a  member  of  the  faculty,  then  the 
university  had  already  made  a  respectable  beginning  in  the  study  of 
exact  science. 

The  year  1856  marks  the  earliest  dawn  of  the  '■  elective  system  "  at 
the  University  of  Michigan.  One  of  the  elective  studies  offered  to  Sen- 
iors in  that  year  was  astronomy.  Professor  Briinnow  lectured  on  this 
subject  to  an  elective  class  of  one — James  G.  Watson.*  With  refer- 
ence to  this  class  Dr.  White  happily  said,  that  "  that  was  the  best 
audience  that  any  professor  in  Michigan  University  ever  had."  Briin- 
now, with  his  pupil  Watson,  reminds  us  of  Gauss,  of  Gottingen,  who 
lectured  at  that  great  university  to  less  than  half  a  dozen  students, 
while  Thibaut,  a  mathematician  of  no  scientific  standing,  presented  the 
elements  of  mathematics  to  audiences  of  hundreds.  ''  If  I  had  the 
choice,"  said  Hankel,  *'  I  should  prefer  being  Gauss  to  Thibaut."  If 
we  had  the  choice,  we  should  prefer  being  a  Briinnow  lecturing  to  one 
or  two  Watsons,  rather  than  being  very  ordinary  teachers  lecturing  to 
large  classes  of  easy-going  students. 

Watson  was  born  in  upper  Canada  in  1838.  He  early  exhibited  ex- 
traordinary mental  power  and  activity.  When  the  lad  was  twelve  his 
parents  were  anxiously  casting  about  to  secure  for  him  the  privileges 
of  a  liberal  education.  They  looked  eastward  to  Toronto  and  westward 
to  Michigan.  Being  in  humble  circumstances,  they  chose  the  latter 
place,  because  education  there  was  free.  Young  Watson  entered  at  the 
Ann  Arbor  High  School,  but  after  an  attendance  of  one  day  and  a  half 
he  was  graduated,  for  it  was  found  that  in  the  sciences  he  was  alto- 
gether beyond  anything  which  his  teachers  had  thought  of.  The  pov- 
erty of  his  parents  made  it  necessary  for  him  to  partly  rely  upon  his 
own  support.  At  this  time  the  future  astronomer  could  be  seen  going 
about  sawing  wood  for  boys  in  college,  while  his  mother  took  in  wash- 
ing to  support  herself  and  boy.  At  the  university  Watson  displayed 
as  much  talent  for  languages  as  he  did  for  mathematics.  The  story 
goes  that  he  decided  between  mathematics  and  Greek,  as  his  specialty, 
by  throwing  a  penny.  "  There  slips  the  penny,  for  which?"  A  notice- 
able exploit  in  the  Junior  year  was  his  reading  the  entire  M^canique 

*  Our  remarks  on  Professor  Wateon  are  drawn  cliiefly  from  an  address  delivered  by 
Prof.  J.  C.  Freeman,  of  the  University  of  Wieconsiu,  aud  printed  in  the  ^gis,  Vol. 
I,  No.  37,  June  24,  1887. 


INFLUX  OF  FRENCH  MATHEMATICS.  247 

Celeste  of  La  Place.  In  the  Senior  year  he  took  the  course  of  lectures 
under  Briinnow,  spoken  of  above. 

While  yet  very  young,  Watson  contributed  numerous  astronomical 
and  mathematical  articles  to  foreign  journals.  He  published  in  1867, 
at  the  age  of  twenty-nine,  his  great  work  on  Theoretical  Astronomy. 
Its  design  appears  from  these  prefatory  words:  ''  Having  carefully  read 
the  works  of  the  great  masters,  my  plan  was  to  prepare  a  complete 
work  on  the  subject,  commencing  with  the  fundamental  principles  of 
dynamics  and  systematically  treating,  from  one  point  of  view,  all  the 
problems  presented."  The  book  gives  a  systematic  derivation  of  the 
formulse  for  calculating  the  geocentric  and  heliocentric  places,  and  de- 
termining orbits,  and  for  computing  special  perturbations,  Including 
also  the  method  of  least  squares,  together  with  a  collection  of  auxiliary 
tables,  etc.  The  work  was  translated  into  continental  languages  and 
became  the  text-book  in  many  observatories  in  Germany,  France,  and 
England. 

When  Briinnow  left  Ann  Arbor,  in  1863,  Watson  became  his  successor. 
V/atson  discovered  a  considerable  number  of  Asteroids.  Twenty-three 
times,  says  Professor  Freeman,  he  knew  the  joy  felt  by 

"  Some  watcher  of  the  skies 
When  a  new  planet  swims  into  his  ken." 

He  was  led  to  believe  that  there  existed  between  Mercury  and  the 
sun  a  planet  hitherto  unknown.  During  his  observation  of  the  eclipse 
in  1878,  at  Denver,  he  caught  sight,  as  he  thought,  of  this  new  planet. 

Watson's  genius  made  the  University  of  Michigan  known  in  scientific 
circles  throughout  the  world.  His  mind  was  pre-eminently  fitted  for 
his  specialty.  With  a  powerful  memory  and  great  mechanical  genius, 
he  combined  the  ability  to  grasp  abstruse  problems  by  a  kind  of  intui- 
tion. He  was  a  man  of  wonderful  activity.  Says  Professor  Freeman : 
"  There  was  a  tireless  energy  in  the  man  that  impressed  every  beholder. 
Some  of  you  recall  the  feeling  you  had  when  Grant  or  Sherman  joined 
the  army  in  the  field,  or  when  you  saw  Sheridan  making  his  last  mile 
from  Winchester  to  Cedar  Creek.  Something  of  the  same  inspiration 
Watson  gave  his  associates." 

During  his  directorship  of  the  observatory,  Watson  generally  deliv- 
ered every  year  to  the  student  community  a  course  of  popular  lectures, 
but  was  otherwise  relieved  from  further  duties  of  giving  instruction, 
excepting  to  pupils  intending  to  make  astronomy  their  specialty.  He 
had  little  patience  with  the  average  boy,  but  his  interest  in  his  special 
students  never  flagged.  He  took  great  pains  to  secure  for  them  suita- 
ble positions.  Old  pupils  of  hi^  may  be  found  holding  responsible  posi- 
tions in  the  U.  S.  Navy,  Patent  Office,  and  Coast  Survey.  His  two 
most  favorite  pupils  were  George  C.  Comstock  and  John  Martin  Schae- 
berly.  Watson  took  the  former  with  him  when  he  left  Ann  Arbor,  in 
1879,  to  take  charge  of  the  Washburn  Observatory  at  the  University 
of  Wisconsin.   Mr,  Schaeberly  remained  at  the  Detroit  Observatory  until 


248  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

1888,  when  he  accepted  a  place  at  the  Lick  Observatory.  He  was  suc- 
ceeded at  Ann  Arbor  by  W.  W.  Campbell.  After  Watson  left  Ann 
Arbor,  Prof.  Mark  W.  Harrington  became  director  of  the  observatory 
there. 

During  his  first  years  after  graduation,  Watson  taught,  besides 
astronomy,  mathematics  and  physics.  Thus,  from  1859  to  1860  he  was 
professor  of  astronomy  and  instructor  in  mathematics ;  from  1860  to 
1863,  instructor  in  physics  and  mathematics.  Other  young  instructors 
in  mathematics  of  this  time  were  W.  P.  Trowbridge,  1856  to  1857,  a 
graduate  of  the  U.  S.  Military  Academy ;  and  John  Emory  Clark,  1857 
to  1859.  Both  of  them  became  connected,  later,  with  Yale  College,  the 
former  as  professor  of  mechanical  engineering,  the  latter  as  professor 
of  mathematics.  These  young  men  did  much,  no  doubt,  to  supply  that 
thoroughness  which  was  wanting  in  the  teaching  of  Professor  Williams, 
the  regular  professor  of  mathematics.  A  beneficial  stimulus  to  the 
study  of  pure  mathematics  was  exerted  by  the  department  of  engineer- 
ing j  for  good  work  in  that  departmeut  was  impossible  without  good 
preliminary  instruction  in  pure  mathematics.  Connected  with  the  de- 
partment of  civil  engineering,  from  1855  to  1857,  was  William  Guy 
Peck,  a  graduate  of  West  Point.  He  was  succeeded  by  De  Yolson 
Wood,  who  had  just  graduated  at  the  Eensselaer  Polytechnic  Institute. 
After  leaving  the  University  of  Michigan,  in  1872,  Wood  became  pro- 
fessor of  mathematics  and  mechanics  in  the  Stevens  Institute  of  Tech- 
nology. He  is  the  author  of  Eesistance  of  Materials,  Eoofs  and  Bridges, 
Elementary  Mechanics,  Analytical  Mechanics,  revised  edition  of  Ma- 
han's  Civil  Engineering,  and  Elements  of  Coordinate  Geometry  (includ- 
ing Cartesian  Geometry,  Quaternions,  and  Modern  Geometry).  Pro- 
fessor Wood's  text-books  contain  numerous  examples  to  be  worked  by 
the  student.  These  books  possess  many  good  features,  and  have  been 
used  quite  extensively  in  our  colleges  and  technical  schools.  Professor 
Wood  has  been  a  very  diligent  contributor  to  a  large  number  of  mathe- 
matical and  scientific  periodicals,  and  has  thereby  done  much  toward 
stimulating  interest  and  activity  in  applied  mathematics. 

The  year  1863  is  marked  in  the  history  of  the  University  of  Michigan 
by  the  departure  of  Briinnow  and  the  arrival  of  Olney.  Prof.  Edward 
Olney  occupied  the  chair  of  mathematics  until  his  death,  in  1887.  He 
was  born  in  Moreau,  Saratoga  County,  N.  Y.,  in  1827.  With  slender 
opportunities  for  early  education,  he  achieved  through  his  own  energy 
distinction  as  a  teacher  and  scholar.  He  began  his  career  as  a  teacher 
in  elementary  schools.  Though  he  had  himself  never  studied  Latin,  he 
began  teaching  it  and  he  kept  ahead  of  his  class,  "  because  he  had  more 
application."  He  thus  educated  himself  in  languages  as  well  as  in 
mathematics.  He  acquired  great  teaching  power,  and  it  is  to  this  that 
his  great  success  is  chiefly  due.  During  the  ten  years  preceding  his 
appointment  at  Ann  Arbor,  he  was  professor  at  Kalamazoo  College, 
Michigan. 


INFLUX  OF  FRENCH  MATHEMATICS.  249 

At  the  Uuiversity  of  Michigan  his  teaching  was  marked  by  great 
thoroughness.  He  was  a  rather  slow  man,  and  took  great  pains  with 
the  poorer  students.  He  had  the  happy  faculty  of  inducing  all  students 
to  perform  faithful  work.  It  is  related  that  the  son  of  a  certain  prom- 
inent Congressman  once  labored  under  the  conceit  that  his  father's  repu- 
tation would  exempt  him  from  the  necessity  of  studying  whenever  he 
felt  disinclined  to  do  so.  Once,  when  being  called  upon  to  recite,  he 
answered,  "  not  prepared."  Professor  Oiney  assured  him  that  the  lesson 
was  easy,  asked  him  to  rise  from  his  seat,  and  then  proceeded,  much  to 
the  amusement  of  the  rest  of  the  class,  to  develop  with  him  the  entire 
lesson  of  the  day  by  asking  him  questions.  In  that  way  was  spent  the 
whole  hour.  The  class  was  made  to  assist  him  in  some  of  the  more  dif- 
ficult points.  The  Congressman's  son  concluded,  on  that  occasion,  that 
it  was,  after  all,  more  agreeable  to  his  feelings  to  prepare  his  mathe- 
matics carefully  in  his  own  room  than  to  expose  his  ignorance  before 
the  whole  class  by  being  kept  reciting  for  a  whole  hour.  At  times  Pro- 
fessor Gluey  eujoyed  joking  at  the  expense  of  those  who  would  not  be 
injured  by  it.  The  result  of  his  teaching  was  a  high  average  standing 
among  students.  The  first  important  step  toward  reaching  good 
results  consisted  in  a  strict  adherence  to  the  requirements  laid  down 
for  admission.  If  a  student  failed  in  his  entrance  examination,  then 
Professor  Olney  took  much  pains  to  see  that  the  deficiencies  would  be 
made  up  under  a  competent  private  teacher  who  was  personally  known 
to  him.  The  rigid  requirements  for  admission  gave  the  mathematical 
department  great  leverage. 

Professor  Olney  was  an  active  promoter  of  various  humanitarian 
enterprises,  and  was  much  interested  in  the  educational  work  of  the 
Baptist  denomination,  of  which  he  was  a  member.  He  was  interested 
in  the  progress  of  Kalamazoo  College  (Baptist)  quite  as  much  as  in 
that  of  Michigan  University.  His  library  is  now  the  property  of  that 
college.  At  the  time  of  his  death  he  was  engaged  in  the  revision  of 
his  series  of  text-books  to  meet  the  increased  demands  of  the  times. 

In  1860,  before  Olney  was  connected  with  the  university,  the  terms 
for  admission  were — to  the  classical  course,  arithmetic,  and  algebra 
through  simple  equations;  to  the  scientific  course,  arithmetic,  algebra 
through  quadratic  equations  and  radicals,  and  the  first  and  third  books 
of  Davies'  Legendre.  In  1864  quadratic  equations  were  added  to  the 
classical  course,  and  to  the  scientific  course  the  fourth  book  of  Legen- 
dre. In  1867  the  requirements  for  the  classical  course  were  raised  so 
as  to  equal  those  in  the  scientific  course,  but  in  the  following  year  quad- 
ratic equations  were  temporarily  withdrawn.  The  fifth  book  of  Legen- 
dre was  added  in  the  scientific  course  in  1869.  In  1870  all  of  Legendre 
was  required,  and  five  books  in  the  classical  course.  In  the  next  year 
arithmetic,  Olney's  Complete  Algebra,  and  Parts  I  and  II  of  Olney's 
Geometry  (including  plane,  solid,  and  spherical  geometry),  were  the 
requirements  in  both  courses.    No  changes  have  been  made  since. 


250  TEACHING   AND   HISTOEY   OF   MATHEMATICS. 

The  college  curriculum  in  1854  was,  for  both  courses,  algebra,  geom- 
etry, trigonometry,  analytical  geometry,  and  calculus.  The  next  year 
calculus  was  withdrawn  from  the  classical  course,  but  was  re  instated 
in  1864,  and  in  18G8  was  made  elective.  In  1878  all  courses  except 
those  for  the  degree  of  B.  L.  (English)  embraced  calculus.  In  1881  the 
B.  L.  course  included  trigonometry.  Since  then  calculus  has  been 
elective  in  all  courses  except  the  scientific.  Analytical  geometry  has 
been  added  to  the  B.  L.  course. 

During  the  last  eight  or  ten  years  the  " university  system"  has  been 
growing  rapidly  at  Ann  Arbor.  Mathematical  studies  of  university 
grade  have  been  offered.  Determinants,  quaternions,  and  modern  ana- 
lytical geometry  were  first  announced  in  1878;  higher  algebra  in  1879; 
synthetic  geometry  and  elliptic  functions  in  1885;  theory  of  functions  in 
1886;  differential  equations  (advanced)  in  1887.  The  calculus  of  varia- 
tions (probably  as  much  as  is  contained  in  Church's  or  Courtenay's 
Calculus)  was  announced  first  in  1866. 

The  text-books  which  have  been  used  at  the  University  of  Michigan, 
at  different  periods,  are  as  follows : 

Algebra. — Davies'  Bourdon,  Ray's — Part  II,  Olney's  University  Algebra,  Newcomb's 
College  Algebra,  Chas.  Smith's  Treatise  on  Algebra,  Salmon's  Higher  Algebra,  Burn- 
side  and  Panton. 

Determinants. — Muir,  Scott,  Dostor,  Peck. 

Geometry. — Davies'  Legendre,  Olney,  Ray. 

Trigonometry. — Davies'  Legendre,  Loomis,  Olney. 

Synthetic  Geometry. — Reye,  Steiner. 

Analytic  Geometry. — Davies,  Loomis,  Church,  Olney,  Peirce's  Curves,  Functions, 
and  Forces,  Chas.  Smith,  Salmon,  Frost,  Aldis,  Whitwortb,  Clebsch. 

Calculus. — Davies,  Church,  Loomis,  Courtenay,  Olney,  Price,  Todhunter,  'William- 
son, Jordan. 

Differential  Equations. — Boole,  Forsyth. 

Calculus  of  Variations. — Todhunter,  Carll. 

Quaternions. — Kelland  »&  Tait,  Hardy,  Tait. 

Elliptic  Functions. — Dur^ge,  Bobek,  Jordan. 

Prof.  G.  C.  Com  stock,  of  the  Washburn  Observatory,  gives  the  fol- 
lowing reminiscences  of  the  mathematical  instruction  at  Ann  Arbor  :* 

"  I  entered  the  University  of  Michigan  in  the  fall  of  1873,  with  a 
preparation  in  mathematics  consisting  of  arithmetic,  elementary  algebra 
through  quadratic  equations  and  including  a  very  hurried  view  of 
logarithms,  and  plane,  solid,  and  spherical  geometry.  The  mathemati- 
cal course  given  in  the  university  at  that  time  comprised,  in  the  Fresh- 
man year,  Olney's  University  Algebra,  inventive  geoinetrj^  (consisting 
of  an  assignment  of  theorems  for  which  the  student  was  expected  to 
find  demonstrations),  and  plane  and  si)herical  trigonometry.  In  the 
Sophomore  year,  general  geometry  and  differential  and  integral  calcu- 
lus. Descriptive  geometry  was  required  of  engineering  students,  and 
was  occasionally  taught  to  others. 

*  Letter  to  the  -writer,  November  6,  1868. 


INFLUX  OF  FEENCH  MATHEMATICS.  251 

"  The  Freshmen  were  taught  by  instructors,  usually  young  men  of  not 
much  experience  ia  teaching,  but  once  a  week  they  (the  students)  went 
up  to  Professor  Olney  for  a  review  of  the  week's  work,  and  these  occa- 
sions were  the  trials  of  a  Freshman's  life.  Olney's  stern  and  rigid  discipline 
had  won  for  him  among  students  the  sobriquet  "  Old  Toughy."  He  was 
not,  however,  a  harsh  man,  and  although  the  students  stood  in  awe  of 
him  I  think  that  he  was  generally  liked  by  them.  One  feature  of  the 
weekly  reviews  may  serve  to  illustrate  his  discipline  and  his  power  of 
enforcing  it.  He  insisted  upon  the  attention  of  each  student  being  given 
to  the  demonstrations  and  explanations  which  the  person  reciting  was 
engaged  upon,  and  given  so  closely  that  the  latter  might  be  stopped  at 
any  point  and  any  other  student  required  to  take  up  the  demonstration 
at  that  point  and  carry  it  on  without  duplicating  anything  which  had 
already  been  given. 

''The  University  Algebra  given  the  Freshman  class  contained  an 
elementary  view  of  infinitesimals,  extending  to  the  differentiation  of 
algebraic  functions  and  the  use  of  Taylor's  formula ;  and  also  a  presen- 
tation of  loci  of  equations,  by  which  the  student  became  familiar  with 
the  geometrical  representation  of  an  equation.  The  Sophomore  thus 
came  to  this  study  of  general  geometry  and  calculus  with  some  prelim- 
inary notions  of  these  subjects.  The  study  of  the  calculus  was  elective, 
but  every  Sophomore  was  required  to  take  an  elementary  course  in  gen- 
eral geometry,  and  to  make  use  here  of  the  principles  of  the  calculus 
which  he  had  learned  as  a  Freshman. 

"  Professor  Olney's  tastes  were  decidedly  geometrical  in  character, 
and  he  constantly  sought  to  translate  analytical  expressions  into  their 
geometrical  equivalents,  and  much  of  his  success  as  a  teacher  is  prob- 
ably due  to  this. 

'^Professor  Beman,  on  the  other  hand,  is  an  analyst,  a  ' lightning 
mathematician '  in  the  student  vernacular,  and,  in  my  day,  the  facility 
with  which  he  handled  mathematical  expressions  dazed  and  discouraged 
the  student,  who  usually  felt  that  he  did  not  get  much  from  Professor 
Beman. 

"The  criticism  which  I  should  now  make  upon  the  mathematical 
teaching  which  I  received,  is  that  little  or  no  attempt  was  made  to  point 
out  the  applications  of  mathematics,  and  to  encourage  the  student  to 
apply  it  to  those  numerous  problems  of  physical  science,  of  engineering, 
and  of  navigation,  which  serve  as  powerful  stimulants  to  the  interest. 
The  student  was  taught  how  to  solve  a  spherical  triangle,  and  how  to 
look  out  logarithms  from  a  table,  but  was  never  required  to  solve  such 
a  triangle  and  obtain  numerical  results. 

"  The  text-books  in  use  were  those  written  by  Professor  Olney,  none 
other  being  employed  even  for  reference.  There  were  no  mathematical 
clubs  or  seminaries,  and  no  facilities  offered  for  the  study  of  mathe- 
matics beyond  the  prescribed  curriculum." 


252  TEACHING   AND    HISTORY    OP   MATHEMATICS. 

Professor  Oluey  is  the  author  of  a  complete  set  of  mathematical 
text-books,  which  have  displaced  the  works  of  Davies,  Loomis,  and 
Kobinsou  iu  many  schools,  both  in  the  Bast  and  in  the  West.  His 
works  are  quite  distinctive  in  the  arrangement  of  subjects,  and  mark  a 
decided  advance  over  the  other  books  just  named.  In  the  explanatory 
notes  added  here  and  there,  in  the  tabular  views  at  the  end  of  chapters, 
in  the  judicious  selection  of  examples,  we  see  the  fruits  of  long  experi- 
ence in  the  class-room.  His  books  exhibit  him  in  the  light  of  a  great 
teacher  rather  than  a  great  mathematician.  He  was  greatly  aided  in 
his  work  by  Professor  Beman,  who  prepared  all  the  "  keys "  to  the 
mathematical  books,  and  did  a  great  deal  of  critical  work.  It  has  been 
stated  that  Professor  Olney  could  never  get  his  publishers  to  print  the 
books  in  the  form  which  seemed  the  most  perfect  to  him.  He  consid- 
ered the  traditional  classification  of  mathematical  subjects  very  defect- 
ive, and  wished  to  write  a  System  of  Mathematics  in  which  he  could 
embody  his  own  ideals  on  this  point.  He  thought,  for  example,  that 
a  considerable  part  of  algebra  should  be  taught  before  taking  up  the  ad- 
vanced parts  of  arithmetic,  such  as  percentage  and  its  applications, 
and  that  plane  geometry  should  precede  mensuration  in  arithmetic. 
By  discarding  the  usual  division  of  mathematics  into  separate  volumes 
on  arithmetic,  algebra,  geometry,  etc.,  and  by  writing  a  system  of  math- 
ematics he  hoped  to  introduce  great  improvements.  The  publishers,  on 
the  other  hand,  preferred  the  traditional  classification,  as  the  books 
would  then  meet  with  larger  sale.  Professor  Olney  was  thus  hampered, 
to  some  extent,  in  the  execution  of  his  ideal  scheme. 

In  his  published  works,  the  science  of  geometry  is  brought  under  two 
great  heads.  Special  or  Elementary  Geometry,  and  General  Geometry. 
The  former  consists  of  four  parts :  The  First  Part  is  an  empirical  geom- 
etry, designed  as  an  introduction,  in  which  the  fundamental  facts  are 
illustrated  but  not  demonstrated.  The  Second  Part  contains  the  ele- 
ments of  demonstrative  geometry,  designed  for  schools  of  lower  grade. 
The  Third  Part  was  written  to  meet  the  special  needs  at  the  University 
of  Michigan.  It  was  studied  in  the  Freshman  class  by  students  who 
had  mastered  the  Second  Part.  The  effort  is  made  here  to  encourage 
original  research.  This  part  contains  also  applications  of  algebra  to 
geometry,  and  an  introduction  to  modern  geometry.  The  Fourth  Part 
consists  of  plane  and  spherical  trigonometry,  treated  geometrically. 
The  old  " line-system"  is  still  retained  here. 

General  Geometry  was  intended  to  be  developed  by  him  in  two  sep- 
arate volumes,  but  only  the  first  was  published.  The  first  treats  of 
plane  loci,  the  second  was  intended  for  loci  in  space.  This  first  vol- 
ume may  be  very  roughly  described  as  covering  the  field  generally  oc- 
cupied by  analytical  geometry  and  calculus.  Olney  favored  the  in- 
finitesimal method,  which  he  used  also  in  his  Elementary  Geometry, 
where  he  permits  the  number  of  sides  of  a  regular  polygon  circum- 
scribed about  a  circle  to  become  "  infinite,"  and  to  coincide  with  the 


INFLUX  OP  FRENCH  MATHEMATICS.  253 

circle.  We  are  glad  that  this  method  is  at  the  present  time  beiug  more 
and  more  eliminated  from  elementary  text-books.  It  is  worthy  of  note 
that  in  his  calculus  Olney  gives  the  elegant  method,  discovered  by 
Prof.  James  C.  Watson,  of  demonstrating  the  rule  for  differentiating  a 
logarithm  without  the  use  of  series. 

In  some  courses  the  subjects  have  been  taught  exclusively  by  lectures, 
but  the  present  tendency  is  to  use  the  best  text-book  available,  and 
supplement  it  with  lectures  as  may  be  found  advisable.  Of  late  years 
a  good  deal  of  attention  has  been  given  to  the  careful  and  critical  read- 
ing of  such  works  as  Salmon's  Conic  Sections,  Higher  Algebra,  Geom- 
etry of  Three  Dimensions,  Frost's  Solid  Geometry,  Jordan's  Gours 
cP Analyse,  Forsyth's  Differential  Equations,  Price's  Calculus,  Carll's  Cal- 
culus of  Yariations,  Burnside  and  Panton's  Theory  of  Equations,  Eeye's 
Geometrie  der  Lage,  Stein er's  Vorlesungen  ilher  synthetiscJie  Geometries 
Olebsch's  Vorlesungen  ilher  Geometrie  der  Ebene.  It  is  thought  that 
better  results  have  been  secured  in  this  way  than  when  the  student's 
attention  is  largely  given  to  the  taking  of  notes. 

Since  the  death  of  Professor  Olney,  Professor  Beman  has  been  filling 
the  professorship  of  mathematics.  He  graduated  at  the  University  of 
Michigan  in  1870.  Excepting  the  first  year  after  graduation  (when  he 
was  instructor  in  Greek  at  another  institution),  he  has  been  teaching 
continually  at  his  alma  mater — from  1871  to  1874  as  instructor  in  math- 
ematics, then  as  assistant  professor  and  as  associate  professor  of  math- 
ematics, and,  finally,  as  full  professor.  He  has  done  much  toward 
introducing  the  ^'  university  system"  in  his  department,  and  has  been  a 
contributor  to  our  mathematical  journals,  particularly  to  the  Analyst 
and  the  Annals  of  Mathematics. 

For  several  years  Charles  1^.  Jones  has  been  professor  of  applied 
mathematics.  He  has  been  a  very  successful  teacher  of  mechanics. 
Professor  Beman  has  two  or  three  assistants  in  the  department  of  pure 
mathematics. 

A  mathematical  club  was  organized  in  1887.  It  is  under  the  control 
of  the  students,  but  an  active  interest  is  continually  shown  by  the 
various  instructors.  Papers  of  some  length  are  presented,  iDroblems 
discussed,  etc. 

UNIVEESITY  OF  WISCONSIN. 

Tlie  University  of  Wisconsin  was  organized  in  1848,  and  formally 
opened  in  1850.  A  preparatory  department  was  established  in  1849,  and 
it  was  not  till  1851  that  regular  college  classes  were  formed.  Like  most 
other  State  universities,  the  University  of  Wisconsin  had  a  hard  strug- 
gle for  existence  during  its  early  years.  Our  Stale  Legislatures  did  not 
always  pursue  a  wise  course  toward  their  higher  institutions  of  learn- 
ing. The  lands  which  were  granted  to  the  States  by  the  General  Gov- 
ernment for  the  support  of  higher  education  were  disposed  of  in  a  man- 
ner intended  to  "  encourage  immigration,"  rather  than  to  foster  a  great 


254  TEACHING   AND    HISTORY   OF   MATHEMATICS. 

university.  Bnt  in  later  years,  say  since  1875,  the  policy  of  the  Wis- 
consin Legislature  has  been  much  more  liberal,  and  the  university  has 
been  advancing  with  prodigious  strides. 

The  first  professor  connected  with  the  institution  was  John  W.  Ster- 
ling. He  was  the  teacher  in  the  preparatory  department,  which  was 
started  in  1849,  in  a  small  building,  before  the  university  had  any  large 
buildings  of  its  own.  After  the  college  department  was  organized. 
Sterling  became  professor  of  mathematics  and  natural  philosophy, which 
position  he  retained  until  about  1867,  when  a  separate  chair  was  created 
for  physics.  From  that  time  on  until  June,  1881,  he  was  professor  of 
mathematics. 

Professor  Sterling  was  born  July  17, 1816,  in  Wyoming  County,  Pa., 
and  died  in  March,  1885,  at  Madison,  Wis.  He  was  graduated  at  the 
College  of  New  Jersey  in  1840,  and  at  the  Princeton  Theological  Semi- 
nary in  1844.  His  mathematical  and  astronomical  instruction  at  Prince- 
ton must  have  been  received  from  Prof.  A.  B.  Dod  and  Prof.  Stephen 
Alexander.  He  went  to  Wisconsin  in  1846,  and  became  professor  in 
Carroll  College,  Waukesha.  Three  years  later  he  entered  upon  his  long 
career  as  professor  at  the  University  of  Wisconsin.  For  one-third  of  a 
century  he  was  connected  with  that  institution.  Never  did  man  work 
more  faithfully  than  did  he  for  its  advancement.  When  the  university 
was  passing  through  the  '•^Sturm  und  Drang  Feriode,''^  and  when  it  was 
without  a  head,  he  more  than  once,  as  dean  of  the  faculty,  assumed  the 
duties  of  president.  He  was  a  man  of  industry  and  energy,  and  was 
ready  to  teach  any  branch,  on  an  emergency.  Among  the  students  he 
was  popular.  He  encouraged  faint-hearted  students,  took  them  to  his 
table,  lent  or  gave  them  money  when  he  had  little  himself.  He  invaria- 
bly treated  students  like  gentlemen  of  mature  judgment  and  common 
sense.  The  great  mass  of  students  appreciated  this,  but  occasionally 
there  were  some  too  young  to  do  so  and  who  should  have  received 
severer  treatment  and  more  summary  action.  In  his  prime  Professor 
Sterling  was  a  man  of  great  physical  strength.  During  even  his  last 
years  he  walked  as  erect  as  a  young  man  of  twenty. 

He  took  a  living  interest  in  mathematics  even  during  the  last  days  of 
his  life.  Though  he  may  not  have  kept  pace  with  recent  advances  in 
this  science,  he  had  a  good  knowledge  of  such  subjects  as  were  treated 
in  our  ordinary  American  text-books.  He  never  published  any  works 
of  his  own.  When  Professor  Watson,  the  astronomer,  came  from  Ann 
Arbor  to  the  University  of  Wisconsin,  in  order  to  take  charge  of  the 
magnificent  new  observatory  erected  by  the  munificence  of  Governor 
Washburn,  an  agreement  was  contemplated  or  reached  between  Wat- 
son and  Sterling  to  prepare  jointly  a  series  of  mathematical  text-books. 
Watson's  wonderful  mathematical  talent  and  Sterling's  long  experience 
in  teaching  would,  indeed,  have  made  a  strong  combination,  but  the 
scheme  was  frustrated  by  the  untimely  death,  in  1880,  of  the  great 
astronomer. 


INFLUX  OF  FRENCH  MATHEMATICS.  255 

In  the  ckiss-room  Sterling's  discipline  was  characterized  bj*  great 
mildness.  He  would  carefully  explain  to  the  class  the  principal  parts 
of  each  lesson.  Even  in  the  last  year  of  his  teaching  his  prelections  were 
always  very  clear,  and  any  student  who  felt  a  desire  to  understand  the 
subjects  which  he  taught,  could  certainly  do  so  by  following  the  exposi- 
tion given  in  the  class.  While  Professor  Sterling  always  explained  well, 
he  was,  in  his  last  years  at  least,  not  sufficiently  exacting;  he  would 
not  compel  a  boy  to  study.  The  consequence  was  that  some  got  from 
him  a  good  knowledge  of  elementary  mathematics,  while  others  took 
advantage  of  the  professor's  leniency.  In  calculus  he  taught  both  the 
method  of  limits  and  the  infinitesimal  method.  The  text-book  was 
based  on  the  former,  but  Professor  Sterling  rather  favored  the  latter. 
The  principles  of  the  calculus  were  not  always  unfolded  with  desired 
rigor,  and  not  uufrequently  some  of  the  best  scholars  in  the  class  shook 
their  heads  at  the  uuceremonial  rejection  of  quantities,  simply  because 
they  were  very,  very  small. 

Among  students  Professor  Sterling  went  by  the  familiar  name  of 
"  Johnnie."  In  1881  he  was  made  professor  emeritus  of  mathematics. 
Though  his  active  duties  in  the  class-room  ceased  at  that  time,  he  con- 
tinued to  take  a  living  interest  in  all  matters  isertaining  to  the  univer- 
sity  to  the  end  of  his  days. 

We  are  not  able  to  give  the  courses  in  mathematics  during  the  early 
days  of  the  university.  During  the  last  years  of  his  teaching  Professor 
Sterling  used  Loomis's  works  throughout.  From  a  communication 
received  by  the  writer  from  Prof.  James  D.  Butler,  it  would  appear  that 
the  works  of  Loomis  were  the  first  ones  taught  in  pure  mathematics  by 
Professor  Sterling.  In  algebra  he  used  Loomis,  afterward  Davies,  and 
then  again  Loomis.  In  conic  sections  he  used  at  one  time  the  work  of 
Cof&n.  " Smith's  Analytic  Geometry"  is  also  one  of  the  books  men- 
tioned. This  was  most  likely  F.  H,  Smith's  translation  of  Biot.  Other 
books  mentioned  are  Peck's  Mechanics,  Robinson's  Astronomy,  Snell's 
Olmsted's  Astronomy,  Snell's  Olmsted  in  optics  and  pneumatics,  and 
Loomis's  Calculus. 

In  1876-77  the  Fresnmen  studied  Loomis's  Algebra,  beginning  with 
quadratics,  Loomis's  Geometry,  Loomis's  Plane  Trigonometry.  The 
Sophomores  were  instructed  in  Loomis's  Conic  Sections  and  Analytic 
Geometry,  Practical  Surveying  (six  weeks),  and  Calculus.  This  ended 
the  course  in  pure  mathematics.  The  Juniors  were  offered  Peck's  Me- 
chanics and  "lectures."  The  mathematical  course  for  engineering  stu- 
dents embraced  also  descriptive  geometry  (Church).  All  students  were 
required  to  pursue  mathematics  through  analytic  geometry;  the  cal- 
culus was  elective  except  to  students  in  civil  and  mechanical  engineer- 
ing. Until  about  the  year  1878,  William  J.  L.  Nicodemus  was  professor 
of  military  science  and  civil  and  mechanical  engineering.  He  was 
spoken  of  by  students  as  a  man  of  great  ability  in  his  line.  On  the 
death  of  Nicodemus  one  of  his  pupils,  Allan  Darst  Conover,  assumed 


256  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

charge  of  the  department  of  civil  engineering.  In  mechanical  engi- 
neering the  instruction  fell  into  the  hands  of  Storm  Bull,  a  relative  of 
the  celebrated  Ole  Bull.  Prof.  Storm  Bull  studied  at  the  Polytechni- 
cum  in  Zurich,  Switzerland,  and  is  a  thorough  master  of  the  subjects 
which  he  teaches.  Descriptive  geometry  has  been  taught  by  him  ever 
since  his  connection  with  the  university. 

On  Sterling's  retirement  the  management  of  the  mathematical  de- 
partment was  entrusted  to  Charles  A.  Van  Yelzer,  a  young  man  who 
for  three  years  had  listened  to  the  inspiring  words  of  Professor  Sylves- 
ter at  the  Johns  Hopkins  University.  Van  Velzer  graduated  at  Cor- 
nell University  in  1876.  After  having  been  instructor  at  his  ahna  mater 
for  one  year,  he  went  to  the  Johns  Hopkins  University,  where  he  was 
honored  with  a  fellowship  in  mathematics.  His  power  for  original  re- 
search is  exhibited  in  his  contributions  to  the  American  Journal  of 
Mathematics  (on  "  Compound  Determinants  "),  the  Johns  Hopkins  Uni- 
versity Circulars,  the  Analyst,  and  the  Mathematical  Magazine. 

In  the  fall  of  1888  appeared,  in  two  separate  volumes,  a  preliminary 
edition  of  Van  Velzer  and  Slichter's  "  Course  in  Algebra."  Slichter  is 
assistant  professor  of  mathematics  at  the  university.  This  book  is 
now  used  in  the  Freshman  class.  The  preliminary  edition  was  gotten 
up  for  the  purpose  of  being  tested  in  the  class-room.  After  the  test, 
such  revisions  will  be  made  as  experience  may  seem  to  require.  In  the 
regular  edition  the  two  parts  will  be  placed  together  in  one  volume. 
The  work  is  not  intended  for  beginners,  but  for  students  entering  the 
Freshman  class  of  our  colleges,  who  already  possess  a  fair  knowledge 
of  the  elements.  It  impresses  the  progressive  teacher  as  being  differ- 
ent from  most  other  works,  and  of  great  excellence.  Many  an  anti- 
quated and  traditional  notion  has  been  thrown  overboard,  and  many 
new  features  have  been  introduced.  They  have  not  been  adopted 
simply  for  the  sake  of  producing  a  book  different  from  others ;  on  the 
contrary,  the  authors  have  profited  by  what  seemed  good  in  other  alge- 
bras. 

The  first  volume  of  Van  Velzer  and  Slichter's  Algebra  embraces,  in 
addition  to  the  usual  subjects,  the  theory  of  limits  and  derivatives. 
In  the  treatment  of  series  the  authors  not  only  state,  but  emphasize  the 
fact  that  infinite  series  must  be  convergent  in  order  to  be  used  with 
safety.  Some  teachers  might  doubt  the  expediency  of  introducing 
Taylor's  Formula  into  a  book  on  algebra  on  account  of  the  difficulty 
encountered  in  a  complete  and  rigorous  proof  of  it. 

The  second  volume  contains  chapters  on  imaginaries,  the  discussion 
of  the  rational  integral  function  of  x^  the  solution  of  numerical  equations 
of  higher  degree,  graphic  representation  of  equations,  and  determinants. 
These  five  chapters  cover  75  pages.  The  treatment  of  these  subjects 
appears  to  us  admirable.  Not  more  is  given  on  each  subject  than  can 
be  conveniently  taught  in  any  college  whose  pupils  possess  a  thorough 
knowledge  of  algebra  through  quadratics  before  entering  the  Freshman 


INFLUX  OF  FRENCH  MATHEMATICS.  257 

class.  A  pleasant  feature  of  the  work  is  the  occasional  "historical 
notes."  This  is  the  first  American  work  on  algebra,  as  far  as  we  know, 
which  states  exjilicitly  that  the  logarithms  invented  by  Napier  are  dif- 
ferent from  the  natural  logarithms. 

The  strongest  feature  of  this  algebra  is  its  style.  Students  who  have 
been  in  Professor  Van  Velzer's  class-room  will  perceive  that  his  great 
power  of  oral  explanation  and  elucidation  has  been  happily  transferred 
to  the  printed  page.  Fowhere  is  the  language  of  the  book  above  the 
comprehension  of  ordinary  students.  The  objective  method  of  explana- 
tion is  adopted  throughout. 

To  Professor  Yan  Yelzer  belongs  the  credit  of  introducing  the  mod- 
ern higher  mathematics  into  the  University  of  Wisconsin.  The  writer 
knows  of  a  student  of  great  taste  for  mathematics,  who  studied  Loomis's 
Calculus  in  the  year  preceding  the  arrival  of  Van  Yelzer,  in  Madison, 
and  who  labored  under  the  impression  that  he  had  mastered  about  all 
that  was  to  be  known  in  pure  mathematics.  He  was  no  little  surprised 
when  the  new  professor,  fresh  from  the  Johns  Hopkins  University,  began 
to  talk  about  determinants,  quaternions,  theory  of  functions,  theory  of 
numbers,  and  multiple  algebra.  The  student's  pride  was  wounded  when 
he  learned  that  Loomis's  Calculus  could  convey  only  a  very  meagre 
knowledge  of  the  transcendental  analysis. 

In  1883  some  alterations  were  made  in  the  mathematical  require- 
ments for  admission.  Daring  the  years  immediately  preceding,  the 
requirements  for  all  the  courses  of  the  university  had  been,  arithmetic, 
algebra  through  quadratic  equations,  and  plane  geometry.  At  the 
time  named  above,  solid  geometry  was  added  to  the  requisitions  for  all. 
regular  courses  in  the  university  except  the  "  ancient  classical." 

The  university  has  established  close  and  friendly  relations  with  the 
high  schools  in  the  State,  and  the  number  of  "  accredited  high  schools" 
is  now  fifty-six.  Of  these  only  six,  however,  prepare  students  for  all 
courses  in  the  university.  This  intimate  relation  with  the  high  schools 
has  had  a  wholesome  influence  upon  both  the  university  and  the  high 
schools. 

As  regards  the  regular  classes  of  mathematics  in  the  college,  we  may 
say  that  since  the  retirement  of  Professor  Sterling,  Loomis's  Algebra 
has  been  retained  until  1888.  Now,  Van  Velzer  and  Slichter's  Algebra 
is  used.  In  trigonometry,  Wheelei's  work  was  introduced  in  1882.  Be- 
fore that  time  Loomis's  was  taught.  In  solid  geometry,  Wentworth's 
has  been  used  lately,  in  place  of  Loomis's.  In  analytical  geometry, 
Loomis's  work  was  superseded  some  years  ago  by  the  English  work  of 
Smith.  In  calculus,  Professor  Van  Velzer  has  taught  Byerly's,  but  this 
year  (1888-89)  he  is  using  Nevi-comb's. 

In  the  Sophomore  year,  and  more  especially  in  the  Junior  and  Senior 

years,  the  elective  system  has  been  in  operation,  with  some  restrictions. 

Since  1881  elective  studies  in  pure  mathematics,  covering  the  calculus 

and  other  branches,  have  been  offered  every  year.    There  have  always 

881— No.  3 — -17 


258  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

been  students  with  a  taste  for  the  higher  branches  of  mathematics.  In 
later  years  the  attendance  upon  these  branches  has  been  on  the  increase. 
In  the  winter  term  of  1881-82  determinants  were  taught  for  the  first 
time  at  the  University  of  Wisconsin.  In  the  spring  term  was  organized 
a  class  of  five  or  six  students  in  quaternions.  Hardy's  text-book  was 
used.  Lately  Professor  Van  Velzer  has  preferred  the  work  by  Kelland 
and  Tait.  During  the  year  1882-83  there  was  an  elective  class  of  about 
the  same  number  as  the  preceding,  studying  Boole's  Differential  Equa- 
tions. Professor  Yan  Yelzer's  constant  aim  was  to  induce  students  to 
do  independent  work.  He  was  always  glad  to  listen  to  such  modified 
treatment  of  the  lesson  in  the  book  as  the  student  might  think  of.  This 
method  of  conducting  the  recitation  gave  rise  to  many  interesting  and 
profitable  discussions.  Considerable  time  was  given  to  the  sabject 
of  singular  solutions.  The  work  of  Boole  was  still  used  in  1887-88, 
but  from  now  on,  that  of  Forsyth  will  be  used,  the  former  being  out  of 
print. 

The  special  courses  in  pure  mathematics  during  the  last  two  years 
have  been  as  follows :  Class  of  two  students  in  Boole's  Differential  Equa- 
tions in  the  fall  term  of  1886-87,  three  hours  per  week ;  class  of  two  in 
the  same  text-book,  winter  term  of  1886-87,  two  hours  per  week ;  class 
of  six  in  modern  algebra  (no  text-book),  winter  term  of  1886-87,  three 
hours  per  week ;  class  of  six  in  Boole's  Differential  Equations,  winter 
term  of  1887-88,  three  hours  per  week ;  class  of  three  in  Boole's  Differ- 
ential Equations,  spring  term  of  1887-88,  three  hours  per  week ;  class 
of  six  in  Kelland  and  Tait's  Quaternions,  spring  term  of  1887-88,  three 
hours  per  week;  class  of  seven  in  Smith's  Analytical  Geometry  of  Three 
Dimensions,  fall  term  of  1887-88,  three  hours  per  week ;  class  of  three 
in  quantics  (Salmon),  fall  term  of  1887-88,  two  hours  per  week. 

It  is  the  practice  at  the  University  of  Wisconsin  to  give  special  honors, 
upon  the  recommendation  of  the  professors  in  the  several  departments, 
to  the  candidates  for  the  bachelor's  degree  who  have  done  special  work 
under  the  direction  of  the  professor  of  any  department  and  prepared 
an  acceptable  thesis;  but  the  amount  of  work  required  for  a  special 
honor  must  be  at  least  the  equivalent  of  a  full  study  for  one  term,  and 
in  case  of  those  branches  in  which  there  are  longer  or  shorter  elective 
courses,  the  student  must  have  taken  the  longer  course.  It  has  been 
specified,  furthermore,  that  candidates  for  special  honors  must  have  a 
general  average  standing  of  85,  and  one  of  93  per  cent,  in  the  de- 
partment of  which  the  application  is  made. 

The  number  of  special-honor  students  in  mathematics  in  late  years 
has  been  quite  as  great,  if  not  greater,  than  in  any  other  department, 
though  the  studies  of  this  department  are,  to  say  the  least,  as  difiBlcult 
as  those  of  any  other.  In  the  class  of  '83  there  were  three  special-honor 
men  in  mathematics.  The  titles  of  their  theses  were  as  follows :  "  Singu- 
lar Solutions  of  Differential  Equations,"  <*Pole  and  Polar  and  Eeciprocal 


INFLUX  OF  FRENCH  MATHEMATICS.  259 

Polars  in  Curves  and  Surfaces  of  the  Second  Order,",  and  "  Development 
and  Dissection  of  Riemann's  Surfaces."  Should  it  be  claimed  that  these 
theses  are  the  work  of  immature  students,  then  we  may  answer  that 
for  candidates  for  the  bachelor's  degree  they  are  nevertheless  creditable. 
The  writer  of  the  first  thesis  (L.  M.  Hoskins)  is  now  doing  excellent  work 
as  instructor  in  engineering  at  the  university.  The  writer  of  the  second 
thesis  (L.  S.  Hulburt)  is  now  professor  of  mathematics  at  the  univer- 
sity of  Dakota.  M.  TJpdegraff,  of  the  class  of  '84,  wrote  a  thesis  on 
"  Eesultants."  He  holds  now  a  responsible  position  at  the  National 
Observatory  at  Cordoba,  Argentine  Republic,  South  America.  Titles 
of  later  "  special-honor  theses  "  are,  ^'Approximation  to  the  Eoots  of 
Numerical  Equations,"  "  Maxima  and  Minima,"  *'  On  the  Equation 
sin  my.  cos  ny  =  sin  mx.  cos  wa?,"  "Different  Systems  of  Co-ordinates." 
These  theses  are  certainly  indicative  of  a  healthful  activity  in  the 
under-graduate  mathematical  department. 

For  special  studies  pursued  after  graduation  and  the  presentation  of 
an  acceptable  thesis,  the  degree  of  Master  is  conferred.  The  following 
are  the  titles  of  two  theses  written  to  secure  the  degree  of  "  master  of 
science  in  mathematics :  "  "  The  Hodograph,"  "  On  a  Quadratic  Form" 
(in  the  theory  of  numbers). 

The  present  courses  in  mathematics  offered  at  the  University  of  Wis- 
consin (catalogue  1887-88)  are  as  follows : 

Sabcourse  I,  Algebra.    Five  exercises  a  .week  during  tlie  fall  term.     (Professor 
Van  Velzer  and  Mr.  Slichter.) 
Required  of  Freshmen  in  all  courses. 

Subcourse  II,  Theory  of  Equations,  including  tbe  elements  of  determinants,  and 
graphic  algebra.    Five  exercises  a  week  during  the  winter  term.     (Professor  Van 
Velzer  and  Mr.  Slichter. ) 
Bequired  of  Freshmen  in  the  Modern  Classical,  English,  General  Science,  and  Engineering 

Courses. 

Subconrse  III,  Solid  Geometry.     Five  exercises  a  week  daring  the  winter  term. 
(Professor  Van  Velzer  or  Mr.  Slichter.) 
Bequired  of  Freshmen  in  the  Ancient  Classical  Course. 

SxibconTselY,  Trigonometry.    Five  exercises  a  week  du.'ing  the  sjiring  term.   (Pro= 
fessor  Van  Velzer  and  Mr.  Slichter.) 
Bequired  of  Freshmen  in  all  courses. 

Subcourse  V,  Descriptive  Geometry.  The  topics  taught  embrace  the  projection  of 
lines,  planes,  surfaces,  and  solids,  the  intersection  of  each  of  these  with  any  one  of 
the  others,  tangent  lines  to  curves  and  surfaces  and  tangent  planes  to  surfaces,  prob- 
lems in  shades  and  shadows,  of  lines  and  surfaces,  linear  perspective  and  isometric 
projection.  The  class-room  exercises  are  accompanied  by  work  in  the  draughting 
room.  The  text-book  used  is  Church's  Descriptive  Geometry.  Full  study  during  the 
spring  term,  Freshman  year,  and  three-fifths  study  during  the  fall  term,  Sophomore 
year.  (Professor  Bull.) 
Bequired  of  Freshmen  in  Civil  and  Mechanical  Engineering.    Elective  for  other  students. 

Subcourse  VI,  Analytic  Geometry.    Five  exercises  a  we6k  during  the  fall  term. 
(Professor  Van  Velzer. ) 
Bequired  of  engineering  Sophomores  and  scientific  Sophomores  who  pursue  mathematical, 

physical,  or  astronomical  studies.    Elective  for  other  students. 


260  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

Subcoarse  VII,  Differential  Calculus.    Five  exercises  a  ■week  during  the  winter 
term.    (Professor  Van  Velzer.) 
' Eequired  of  engineering  Sopliomores  and  scientific  Sophomores  xoho  pursue  mathematical, 

physical,  or  astronomical  studies.    Elective  for  other  students. 

Subcourse  VIII,  Integral  Calculus.    Five  exercises  a  week  during  tbe  spring  term. 
(Professor  Van  Velzer.) 
Eequired  of  engineering  Sophomores  and  scientific  Sophomores  who  pursue  mathematical, 

physical,  or  astronomical  studies.    Elective  for  other  students. 

Subcourse  XIX,  Method  of  Least  Squares.  This  is  a  course  in  the  theory  of  probabil- 
ities as  applied  to  the  adjustment  of  errors  of  observation.    It  will  be  first  given  in 
1889.    Must  be  preceded  by  eubcourses  VI,  VII,  and  VIII,  three-fifths  study  during 
the  winter  term.     (Mr.  Hoskins.) 
acquired  of  Seniors  in  Civil  Engineering. 

Subcourses  IX  to  XVIII,  special  advanced  electives.  Courses  varying  from  year  to 
year  are  offered  in  the  following  subjects :  IX,  Modern  Analytic  Geometry  ;  X,  Higher 
Plane  Curves ;  XI,  Geometry  of  Three  Dimensions;  XII,  Differential  [Equations ;  XIII, 
Spherical  Harmonics  ;  XTV ,  Elliptic  Functions  ;  XY ,  Theory  of  Functions ;  XVI,  Theory 
of  Numhers  ;  XVII,  Quantics ;  anc?  XVIII,  Quatei'nions. 

Very  good  work  has  been  done,  at  times,  by  students  in  the  depart- 
ment of  mathematical  physics.  Prof.  John  E.  Davies,  the  professor  of 
physics,  takes  a  living  interest  in  pure  as  well  as  applied  mathematics. 
His  reading  in  pure  mathematics  has,  indeed,  been  very  extensive. 
Mathematical  reading  is  a  recreation  to  him.  He  would  not  uufre- 
quently  take  with  him  some  mathematical  work— as,  for  instance,  Tait's 
Quaternions— to  faculty  meetings,  that  he  might  pass  pleasantly  the 
otherwise  tedious  sessions  of  that  august  assembly.  Many  years  ago  he 
made,  for  his  own  use,  a  complete  translation  of  Koenigsberger's  work 
on  Elliptic  Functions. 

The  university  offers  excellent  facilities  for  the  study  of  astronomy. 
The  Washburn  Observatory  has  a  large  equatorial  for  use  in  original 
work,  and  also  a  smaller  one  for  the  use  of  students.  After  the  death  of 
Professor  Watson,  Professor  Holden  became  director  of  the  Observatory. 
He  held  this  position  until  his  appointment  as  director  of  the  Lick 
Observatory.  Prof.  George  0.  Comstock  is  now  professor  of  astron- 
omy and  associate  director  of  the  Washburn  Observatory.  Professor 
Comstock  is  a  pupil  of  Watson,  and  came  to  Wisconsin  from  Ann 
Arbor  with  Watson.  Before  assuming  the  duties  of  his  present  posi- 
tion he  was  for  two  or  three  years  x)rofessor  of  mathematics  and  astron- 
omy at  the  University  of  Ohio. 

The  instruction  in  analytical  mechanics  is  in  charge  of  Mr.  L.  M. 
Hoskins,  a  young  man  of  very  marked  mathematical  talent.  He  grad- 
uated in  1883  at  the  head  of  a  class  of  sixty-five,  and  was  afterward 
appointed  fellow  in  mathematics  in  Harvard  University.  Through  his 
influence,  the  study  of  analytical  mechanics  had  been  made  much  more 
l)rominent  in  the  engineering  courses  than  it  had  been  formerly.  Two 
terms  are  now  devoted  to  it  instead  of  only  one.  Bowser's  Elements 
of  Analytical  Mechanics  is  the  text-book  used. 

Mr.  Hoskins  has  contributed  to  the  Annals  of  Mathematics,  the 
Mathematical  Magazine,  and  Yan  Nostrand's  Engineering,  Magazine. 


INFLUX    OF   FEENCH    MATHEMATICS.  261 

JOHNS  HOPKINS  UNIVERSITY, 

President  Daniel  0.  Gilman  once  said  to  the  trustees  of  the  Johns 
Hopkins  University,  when  the  question  of  "  How  to  begin  a  university  " 
was  upon  their  minds,  "  Enlist  a  great  mathematician  and  a  distin- 
guished Grecian ;  your  problem  will  be  solved.  Such  men  can  teach  in, 
a  dwelling-house  as  well  as  in  a  ijalace.  Part  of  the  apparatus  they 
will  bring,  part  we  will  furnish.  Other  teachers  will  follow  them."*  So 
it  came  to  pass  that,  before  there  were  any  buildings  for  classes,  a  pro- 
fessor of  mathematics  and  a  professor  of  Greek  were  secured  for  the 
new  university. 

When  President  Gilman  was  engaged  in  the  all-important  work  of 
selecting  men  for  the  above  positions,  he  may  have  been  actuated  in  his 
choice  by  thoughts  similar  to  those  of  Prof.  G.  Ghrystal,  who,  before  a 
learned  body  of  English  scientists,  once  expressed  himself  as  follows  :t 
"  Science  can  not  live  among  the  people,  and  scientific  education  can  not 
be  more  than  a  wordy  rehearsal  of  dead  text-books,  unless  we  have  living 
contact  with  the  working  minds  of  living  men.  It  takes  the  hand  of  God 
to  make  a  great  mind,  but  contact  with  a  great  mind  will  make  a  little 
mind  greater.  The  most  valuable  instruction  in  any  art  or  science  is  to 
sit  at  the  feet  of  a  master,  and  the  next  best,  to  have  contact  of  another 
who  has  himself  been  so  instructed." 

Is  there  a  student  among  us  who  has  studied  with  Sylvester  and  who 
will  deny  the  truth  of  the  above  ?  Is  there  a  mathematician,  who  has 
sat  as  a  i)upil  at  the  feet  of  Benjamin  Peirce,  who  will  deny  it  ?  It  is  a 
fortunate  circumstance  for  the  progress  of  the  exact  sciences  in  this 
country  that,  at  a  time  when  the  "  Father  of  American  Mathematics  " 
was  ajjproaching  his  grave,  there  came  among  us  another  master  who 
gave  the  study  of  mathematics  a  fresh  and  powerful  impulse.  Profes- 
sor Sylvester  is  a  mathematical  genius,  who  has  no  superior  in  Eng- 
land, except,  perhaps.  Professor  Cayley. 

James  Joseph  Sylvester  was  born  in  London  in  1814,  and  was  edu- 
cated at  the  University  of  Cambridge.  He  came  to  this  country  to  fill 
the  professorship  at  the  University  of  Virginia  when  he  was  a  very 
young  man,  but  his  stay  among  us  then  was  very  short.  He  became  a 
member  of  the  Koyal  Society  at  the  age  of  twenty-five.  For  some  time 
he  was  professor  of  natural  philosophy  in  University  College,  London. 
In  1855  he  became  professor  of  mathematics  in  the  Koyal  Military 
Academy  at  Woolwich,  and  in  1876  was  elected  for  the  position  at  the 
Johns  Hopkins  University. 

Sylvester's  activity  has  been  wonderful.  Prior  to  1863  he  published 
112  scientific  memoirs,  which  are  recorded  in  the  Eoyal  Society's  Index 
of  Scientific  Papers.    A  most  important  paper,  printed  in  the  Philo- 

*  Annual  report  of  the  president  of  the  Johns  Hopkins  University,  1888,  p.  29. 
\Nature,  September  10,  1885,  Section  A  of  Brit.  Assocatiau,  opening  address  hy 
Prof.  G.  Chrystal,  president  of  the  section. 


262  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

sophical  Transactions  of  1864,  is  Sylvester's  Theorem  on  Newton's  Eule 
for  discovering  the  number  of  real  and  imaginary  roots  of  an  equation. 
Of  this  Todhunter  says :  *  "If  we  consider  the  intrinsic  beauty  of  the 
theorem,    *     *    *     the  interest  which  belongs  to  the  rule  associated 
with  the  great  name  of  Newton,  and  the  long  lapse  of  years  during 
which  the  reason  and  extent  of  that  rule  remained  undiscovered  by 
mathematicians,  among  whom  Maclaurin,  Waring,  and  Euler  are  ex- 
plicitly included,  we  must  regard  Professor  Sylvester's  investigations 
made  to  the  theory  of  equations  in  modern  times  justly  to  be  ranked 
with  those  of  Fourier,  Sturm,  and  Cauchy."    A  few  of  his  numerous 
other  investigations,  made  before  coming  to  Baltimore,  are  on  the  Rota- 
tion of  a  Rigid  Body ;  on  the  Analytical  Development  of  Fresnel's 
Optical  Theory  of  Crystals ;  on  Reversion  of  Series ;  on  the  Involution 
of  Six  Lines  in  Space,  "  culminating  in  the  result  that  if  these  six  lines 
represent  forces  in  equilibrium  they  must  lie  on  a  ruled  cubic  surface;'' 
on  a  general  theorem  by  which,  for  instance,  the  quintic  can  be  ex- 
pressed as  the  sum  of  three  fifth  powers.     In  1859  he  gave  a  course  of 
lectures  at  King's  College,  London,  on  the  subject  of  The  Partitions  of 
Numbers  and  the  Solution  of  Simultaneous  Equations  in  Integers,  in 
which  it  fell  to  his  lot  "  to  show  how  the  difficulties  might  be  overcome 
which  had  previously  baffled  the  efforts  of  mathematicians,  and  espe- 
cially of  one  bearing  no  less  venerable  a  name  than  that  of  Leonard 
Euler,"  and  also  laid  the  basis  of  a  method  which  has  since  been  carried 
out  to  a  much  greater  extent  in  his  "  Constructive  Theory  of  Partitions," 
published  in  the  American  Journal  of  Mathematics,  in  writing  which 
he  "  received  much  valuable  co-operation  and  material  contributions" 
from  his  "  pupils  in  the  Johns  Hopkins  University."  t 

Professor  Sylvester's  most  celebrated  work  has  been  in  modern  higher 
algebra.  A  very  large  portion  of  the  theory  of  determinants  is  due  to 
him,  and  the  epoch-making  theory  of  invariants  owes  its  origin  and 
early  development  almost  exclusively  to  his  genius  and  that  of  Pro- 
fessor Cayley. 

The  Johns  Hopkins  University  offered  to  Professor  Sylvester  every 
facility  for  original  work  that  could  be  desired.  By  the  system  ol 
"  fellowsViips "  a  number  of  talented  young  men  were  drawn  to  Balti- 
more, who  were  capable  not  only  of  understanding  the  teachings  ol 
their  great  master,  but,  in  many  cases,  also  of  aiding  him  in  his  re- 
searches. The  university,  moreover,  started  the  American  Journal  ol 
Mathematics,  in  which  all  investigations  in  mathematics  could  be  pub- 
lished and  thereby  brought  before  the  mathematical  public.  Professor 
Sylvester's  time  was  not  taken  up  by  the  usual  routine  work  in  school, 
but  was  almost  wholly  given  to  the  pursuit  of  his  favorite  subjects. 
He  lectured,  perhaps,  two  or  three  times  per  week,  but  these  lectures 
generally  disclosed  some  new  discovery  in  algebra. 

*  Theory  of  Equatio'as,  page  250. 

t  Inaagural  Lecture  delivered  by  Professor  Sylvester  before  tbe  University  of  Ox- 
ford, December  12,  18f35,  published  in  Nature,  January  7,  1886. 


INFLUX  OF  FEENCH  MATHEMATICS.  263 

Though  he  had  passed  his  sixtieth  year  before  he  came  to  the  Johns 
Hopkins  University,  his  mind  seemed  to  be  as  strong  and  active  as 
ever.  The  group  of  students  he  had  gathered  about  him  were  almost 
constantly  made  to  feel  the  glow  of  new  ideas  or  of  old  ones  in  a  new 
form.  From  1877  to  1882,  Professor  Sylvester  contributed  thirty  arti- 
cles and  notes  to  the  American  Journal  of  Mathematics ;  twenty-two  to 
the  Gomptes  Bendus  de  VAcademie  des  Sciences  de  Vlnstitut  de  France  ; 
one  paper  to  the  Proceedings  of  the  Eoyal  Society,  "On  the  Limits  to 
the  Order  and  Degree  of  the  Fundamental  Invariants  of  Binary  Quan- 
tics"  (1878);  four  to  the  Messenger  of  Mathematics;  four  to  the  Lon- 
don,  Edinburgh,  and  Dublin  Philosophical  Magazine;  six  to  the  Journal 
fur  reine  und  angewandte  MathematiJc,  Berlin.*  If  this  list  be  complete, 
the  number  of  original  papers  published  by  him  while  at  the  Johns 
Hopkins  University  was  sixty-seven.  Special  mention  may  be  made 
here  of  a  proof  by  Professor  Sylvester,  printed  in  the  Philosophical 
Magazine  for  1878,  of  a  theorem  on  the  number  of  linearly  independent 
differentiants,  which  had  been  awaiting  proof  for  over  a  quarter  of  a 
century.  He  was  led  to  undertake  the  investigation  of  this  subject  by 
a  question  put  to  him  by  one  of  his  students  in  connection  with  a  foot- 
note given  at  one  place  in  Fa^  de  Bruno's  ThSorie  des  Formes  Binaires. 

Since  his  return  to  England,  Sylvester  has  been  developing  a  new 
subject,  which  he  calls  the  "  Method  of  Reciprocants."  The  lectures 
which  he  delivered  on  this  subject  at  the  University  of  Oxford  have 
been  reported  by  Mr.  Hammond  and  published  in  the  American  Jour- 
nal of  Mathematics. 

Sylvester  has  manufactured  a  large  number  of  technical  terms  in 
mathematics.  He  himself  speaks  on  this  point  as  follows :  "  Perhaps 
I  may,  without  immodesty,  lay  claim  to  the  appellation  of  the  mathe- 
matical Adam,  as  I  believe  that  I  have  given  more  names  (passed  into 
general  circulation)  to  the  creatures  of  the  mathematical  reason  than 
all  the  other  mathematicians  of  the  age  combined."! 

In  his  writings,  Professor  Sylvester  is  often  very  eloquent.  His  style 
is  peculiarly  flowery,  and  indicative  of  very  powerful  imagination. 
His  articles  are  frequently  interspersed  with  short  pieces  of  poetry, 
either  quoted  or  of  his  own  composition.  Thus,  in  his  article  in  Nature, 
January,  1886,  is  given  a  short  poem,  "On  a  Missing  Member  of  a 
Family  Group  of  Terms  in  an  Algebraical  Formula ;"  followed  by  this 
sentence:  "Having  now  refreshed  ourselves  and  bathed  the  tips  of 
our  fingers  in  the  Pierian  spring,  let  us  turn  back  for  a  few  brief  mo- 
ments to  a  light  banquet  of  the  reason." 

Since  the  beginning  of  the  Johns  Hopkins  University,  twenty  fel- 
lowships have  been  open  annually  to  competition,  each  yielding  five 
hundred  dollars  and  exempting  the  holder  from  all  charges  for  tuition. 
This  system  was  instituted  for  the  purpose  of  affording  to  young  men 

*  U.  S.  Bureau  of  Education,  Circular  of  Information  No.  1,  1888,  p.  220, 
t  Nature,  Dec.  15,  1887,  p.  152,  note. 


264  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

of  talent  an  opportunity  of  continuing  their  studies  in  the  university, 
■while  looking  forward  to  positions  as  professors,  teachers,  and  investi- 
gators. They  have  been  given  to  graduate  students  who  showed  partic- 
ular aptitude  for  advanced  work  in  their  chosen  specialty.  During 
the  time  when  Sylvester  was  connected  with  the  university  there  were 
nearly  always  three  or  four  fellowships  granted  to  mathematical  stu- 
dents, but,  in  recent  years  the  number  has  been  reduced  to  two,  in  con- 
sequence of  an  increase  in  the  number  of  departments  in  the  university, 
among  which  the  fellowships  must  be  divided.  Among  the  first  holders 
of  fellowships  in  mathematics  were  Thomas  Craig  (1876-78),  George  B. 
Halsted  (1876-78),  Fabian  Franklin  (1877-79),  W.  I.  Stringham  (1878- 
80),  0.  A.  Van  Velzer  (1878-81),  all  holding  leading  and.  responsible 
positions  now,  as  professors  of  mathematics. 

Professor  Sylvester's  first  high  class  at  the  new  unhversity  consisted  of 
only  one  student,  G.  B.  Halsted,  who  had  persisted  in  urging  Sylvester 
to  lecture  on  the  modern  algebra.  The  attempt  to  lecture  on  this  sub- 
ject led  him  into  new  investigations  in  quantics.  In  his  address  on 
Commemoration  Day  at  the  Johns  Hopkins,  he  spoke  about  this  work 
as  follows : 

"This  is  the  kind  of  investigation  in  which  I  have  for  the  last  month 
or  two  been  immersed,  and  which  I  entertain  great  hopes  of  bringing  to 
a  successful  issue.  Why  do  I  mention  it  here  ?  It  is  to  illustrate  my 
opinion  as  to  the  invaluable  aid  of  teaching  to  the  teacher,  in  throwing 
him  back  upon  his  own  thoughts  and  leading  him  to  evolve  new  re- 
sults from  ideas  that  would  have  otherwise  remained  passive  or  dormant 
in  his  mind. 

"But  for  the  persistence  of  a  student  of  this  university  in  urging 
upon  me  his  desire  to  study  with  me  the  modern  algebra  I  should  never 
have  been  led  into  this  investigation ;  and  the  new  facts  and  principles 
which  I  have  discovered  in  regard  to  it  (important  facts,  I  believe), 
would,  so  far  as  I  am  concerned,  have  remained  still  hidden  in  the  womb 
of  time.  In  vain  I  represented  to  this  inquisitive  student  that  he  would 
do  better  to  take  up  some  other  subject  lying  less  off  the  beaten  track 
of  study,  such  as  the  higher  parts  of  the  calculus  or  elliptic  functions,  or 
the  theory  of  substitutions,  or  I  wot  not  what  besides.  He  stuck  with 
perfect  respectfulness,  but  with  invincible  pertinacity,  to  his  point.  He 
would  have  the  new  algebra  (Heaven  knows  where  he  had  heard  about 
it,  for  it  is  almost  unknown  in  this  continent),  that  or  nothing.  I  was 
obliged  to  yield,  and  what  was  the  consequence  ?  In  trying  to  throw 
light  upon  an  obscure  explanation  in  our  text-book,  my  brain  took  fire, 
I  plunged  with  re-quickened  zeal  into  a  subject  which  I  had  for  years 
abandoned,  and  found  food  for  thoughts  which  have  engaged  my  atten- 
tion for  a  considerable  time  past,  and  will  probably  occupy  all  my 
powers  of  contemplation  advantageously  for  several  months  to  come." 

This  extract  describes  the  beginning  of  his  scientific  activity  and  pro- 
ductiveness in  the  New  World, 


INFLUX  OF  FEENCH  MATHEMATICS.  265 

It  may  not  be  without  interest  to  learn  what  some  of  his  former  i)upils 
at  the  Johns  Hopkins  University  have  to  say  about  him.  Says  Dr. 
G.  B.  Halsted : 

"  Young  Americans  could  hardly  realize  that  the  great  Sylvester, 
who  with  Cayley  outranks  all  English-speaking  mathematicians,  was 
actually  at  work  in  our  land.  All  young  men  who  felt  within  themselves 
the  divine  longing  of  creative  power  hastened  to  Baltimore,  made  at  once 
by  this  Euclid  a  new  Alexandria.  It  was  this  great  awakening  and 
concentration  of  mathematical  promise,  and  the  subsequent  facilities 
offered  for  publication  of  original  work,  which,  rather  than  any  teaching, 
made  the  American  renaissance  in  mathematics.    *    *    * 

"  A  short,  broad  man  of  tremendous  vitality,  the  physical  type  of 
Hereward,  the  Last  of  the  English,  and  his  brother-in-arms.  Winter, 
Sylvester's  capacious  head  was  ever  lost  in  the  highest  cloud-lands  of 
pure  mathematics.  Often  in  the  dead  of  night  he  would  get  his  favor- 
ite pupil,  that  he  might  communicate  the  very  last  product  of  his  cre- 
ative thought.  Everything  he  saw  suggested  to  him  something  new 
in  the  higher  algebra.  This  transmutation  of  everything  into  new  math- 
ematics was  a  revelation  to  those  who  knew  him  intimately.  They 
began  to  do  it  themselves.  His  ease  and  fertility  of  invention  proved 
a  constant  encouragement,  while  his  contempt  for  provincial  stupidities, 
such  as  the  American  hieroglyphics  for  n  and  e,  which  have  even  found 
their  way  into  Webster's  Dictionary,  made  each  young  worker  apply  to 
himself  the  strictest  tests. 

"  To  know  him  was  to  know  one  of  the  historic  figures  of  all  time, 
one  of  the  immortals ;  and  when  he  was  really  moved  to  speak,  his 
eloquence  equalled  his  genius.  I  never  saw  a  more  astonished  man 
than  James  Eussell  Lowell  listening  to  the  impassioned  oratory  of  Syl- 
vester's address  upon  the  bigotry  of  Christians. 

"  That  the  presence  of  such  a  man  in  America  was  epoch-making  is 
not  to  be  worjdered  at.    His  loss  to  us  was  a  national  misfortune."* 

In  answer  to  an  inquiry  about  Sylvester's  methods  of  teaching,  Dr« 
E.  W.  Davis  (fellow  from  1882  to  1884)  writes  hurriedly  as  follows : 
"  Sylvester's  metJiodft !  He  had  none.  '  Three  lectures  w  ill  be  delivered 
on  a  Kew  Universal  Algebra,'  he  would  say;  then,  '  The  course  must  be 
extended  to  twelve.'  It  did  last  all  the  rest  of  that  year.  The  following 
year  the  course  was  to  be  Substitufyions-Theorie,  by  Netto.  We  all  got 
the  text.  He  lectured  about  three  times,  following  the  text  closely 
and  stopping  sharp  at  the  end  of  the  hour.  Then  he  began  to  think 
about  matrices  again.  '  I  must  give  one  lecture  a  week  on  those,'  he 
said.  He  could  not  confine  himself  to  the  hour,  nor  to  the  one  lecture  a 
week.  Two  weeks  were  passed,  and  B"etto  was  forgotten  entirely  and 
never  mentioned  again.  Statements  like  the  following  were  not  unfre- 
quent  in  his  lectures :  '  I  haven't  proved  this,  but  I  am  as  sure  as  I  can 


*  Letter  to  tlie  writer,  December  25,  1888. 


266  TEACHING   AND    HISTOEY    OF    MATHEMATICS. 

be  of  anything  that  it  must  be  so.  From  this  it  will  follow,  etc'  At  the 
next  lecture  it  turned  out  that  what  he  was  so  sure  of  was  false.  Never 
mind,  he  kept  on  forever  guessing  and  trying,  and  presently  a  wonder- 
ful discovery  followed,  then  another  and  another.  Afterward  he  would 
go  back  and  work  it  all  over  again,  and  surprise  us  with  all  sorts  of 
side  lights.  He  then  made  another  leap  in  the  dark,  more  treasures 
were  discovered,  and  so  on  forever." 

Letus  now  listen  to  another  of  his  old  pupils,  Mr.  A.  S.  Hathaway 
(fellow  from  1882  to  1884) : 

"  I  can  see  him  now,  with  his  white  beard  and  few  locks  of  gray  hair, 
his  forehead  wrinkled  o'er  with  thoughts,  writing  rapidly  his  figures  and 
formulas  on  the  board,  sometimes  explaining  as  he  wrote,  while  we,  his 
listeners,  caught  the  reflected  soundsfrom  the  board.  But  stop,  some- 
thing is  not  right,  he  pauses,  his  hand  goes  to  his  forehead  to  help  his 
thought,  he  goes  over  the  work  again,  emphasizes  the  leading  points,  and 
finally  discovers  his  difSculty.  Perhaps  it  is  some  error  in  his  figures, 
perhaps  an  oversight  in  the  reasoning.  Sometimes,  however,  the  diffi- 
culty is  not  elucidated,  and  then  there  is  not  much  to  the  rest  of  the  lect- 
ure. But  at  the  next  lecture  we  would  hear  of  some  new  discovery  that 
was  the  outcome  of  that  difficulty,  and  of  some  article  for  the  Journal, 
which  he  had  begun.  If  a  text-book  had  been  taken  up  at  the  beginning, 
with  the  intention  of  following  it,  that  t^xt-book  was  most  likely  doomed 
to  oblivion  for  the  rest  of  the  term,  or  until  the  class  had  been  made  lis- 
teners to  every  new  thought  and  principle  that  had  sprung  from  the 
laboratory  of  his  mind,  in  consequence  of  that  first  difficulty.  Other 
difficulties  would  soon  appear,  so  that  no  text- book  could  last  more  than 
half  of  the  term.  In  this  way  his  class  listened  to  almost  all  of  the 
work  that  subsequently  appeared  in  the  Journal.  It  seemed  to  be  the 
quality  of  his  mind  that  he  must  adhere  to  one  subject.  He  would  think 
about  it,  talk  about  it  to  his  class,  and  finally  write  about  it  for  the 
Journal.  The  merest  accident  might  start  him,  but  once  started,  every 
moment,  every  thought  was  given  to  it,  and,  as  much  as  possible,  he  read 
what  others  had  done  in  the  same  direction ;  but  this  last  seemed  to  be 
his  weak  point;  he  could  not  read  without  meeting  difficulties  in  the  way 
of  understanding  the  author.  Thus,  often  his  own  work  reproduced 
what  others  had  done,  and  he  did  not  find  it  out  until  too  late. 

"A  notable  example  of  this  is  his  theory  of  cyclotomic  functions, 
which  he  had  reproduced  in  several  foreign  journals,  only  to  find  that 
he  had  been  greatly  anticipated  by  foreign  authors.  It  was  manifest, 
one  of  the  critics  said,  that  the  learned  professor  had  not  read  Rum- 
mer's elementary  results  in  the  theory  of  ideal  primes.  Yet  Professor 
Smith's  report  on  the  theory  of  numbers,  which  contained  a  full  synopsis 
of  Rummer's  theory,  was  Professor  Sylvester's  constant  companion. 

"  This  weakness  of  Professor  Sylvester,  in  not  being  able  to  read  what 
others  had  done,  is  perhaps  a  concomitant  of  his  jieculiar  genius.  Other 
minds  could  pass  over  little  difficulties  and  not  be  troubled  by  them, 


INFLUX  OF  FRENCH  MATHEMATICS.  267 

and  so  go  on  to  a  final  understanding  of  the  results  of  the  author.  But 
not  so  with  him.  A  difficulty,  however  small,  worried  him,  and  he  was 
sure  to  have  difficulties  until  the  subject  had  been  worked  over  in  his 
own  way,  to  correspond  with  his  own  mode  of  thought.  To  read  the 
work  of  others,  meant  therefore  to  him  an  almost  independent  de- 
velopment of  it.  Like  the  man  whose  pleasure  in  life  is  to  pioneer  the 
way  for  society  into  the  forests,  his  rugged  mind  could  derive  satisfac- 
tion only  in  hewing  out  its  own  paths ;  and  only  when  his  efforts  brought 
him  into  the  uncleared  fields  of  mathematics  did  he  find  his  place  in  the 
Universe." 

These  reminiscences  are  extremely  interesting,  inasmuch  as  they  show 
the  workings  of  a  great  mind.  The  mathematical  reader  will  surely 
enjoy  the  following  reminiscences  of  "  Silly,"  by  one  of  his  favorite  pu- 
pils, Dr.  W.  P.  Durfee,  professor  of  mathematics  at  Hobart  College, 
Geneva,  N.  Y.  He  was  a  fellow  in  mathematics  from  1881  to  1883. 
Speaking  of  his  recollections  of  Sylvester,  he  says : 

"  I  don't  know  that  I  can  do  better  than  preface  them  by  an  account, 
as  far  as  my  memory  serves  me,  of  the  work  we  did  while  I  was  at  the 
Johns  Hopkins  University.  I  say  we^  as  I  always  think  of  the  whole 
staff  as  working  together,  so  thoroughly  did  Sylvester  inspire  us  all 
with  the  subject  which  was  immediately  interesting  him.  I  went  to 
Baltimore  in  October,  1881,  as  a  fellow,  and,  though  my  previous  math- 
ematical training  had  been  of  the  scantiest,  I  had  the  courage  of  igno- 
rance and  immediately  began  to  attend  Sylvester's  lectures,  while  Mr. 
Davis  and  some  others  thought  they  would  wait  for  a  year  and  prepare 
themselves  to  profit  by  them.  Sylvester  began  to  lecture  on  the  Theory 
of  Numbers,  and  promised  to  follow  Lejeune  Dirichlet's  book;  he  did 
so  for,  perhaps,  six  or  eight  lectures,  when  some  discussion  which  came 
up  led  him  off,  and  he  interpolated  lectures  on  the  subject  of  frequency, 
and  after  some  weeks  interpolated  something  else  in  the  midst  of  these. 
After  some  further  interpolations  he  was  led  to  the  consideration  of  his 
Universal  Algebra,  and  never  finished  any  of  the  previous  subjects. 
This  finished  the  first  year,  and,  although  we  had  not  received  a  sys- 
tematic course  of  lectures  on  any  subject,  we  had  been  led  to  take  a  liv- 
ing interest  in  several  subjects,  and,  to  my  mind,  were  greatly  gainers 
thereby.  The  second  year,  1882-83,  he  started  off  on  the  subject  of 
substitutions,  but  our  experience  was  similar  to  that  of  the  preceding 
year,  and  I  can  not  now,  after  the  six  years  which  have  intervened,  dis- 
entangle the  various  topics  that  engaged  his  attention.  Amongst  others 
were  Turey's  series,  partitions,  and  universal  algebra.  He  could  not 
lecture  on  a  subject  which  was  not  at  the  same  time  engaging  his  atten-. 
tion.  His  lectures  were  generally  the  result  of  his  thought  for  the  pre- 
ceding day  or  two,  and  often  were  suggested  by  ideas  that  came  to  him 
while  talking.  The  one  great  advantage  that  this  method  had  for  his 
students  was  that  everything  was  fresh,  and  we  saw,  as  it  were,  the 
very  genesis  of  his  ideas.    One  could  not  help  being  inspired  by  such 


268  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

teaching,  and  many  of  us  were  led  to  investigate  on  lines  which  he 
touched  upon.  He  was  always  pleased  at  what  one  had  to  suggest,  and 
generally  bore  interruptions  with  i^atience.  He  would  often  stop  to 
discuss  points  that  arose,  and  accepted  our  opinions  as  of  some  worth. 
I  must  qualify  these  latter  statements  somewhat,  as  he  was  apt  to  be 
partial,  and  it  made  all  the  difference  in  the  world  who  it  was  that  in- 
terrupted him. 

"His  manner  of  lecturing  was  highly  rhetorical  and  elocutionary. 
When  about  to  enunciate  an  important  or  remarkable  statement  he 
would  draw  himself  up  till  he  stood  on  the  very  tips  of  his  toes,  and  in 
deep  tones  thunder  out  his  sentences.  He  preached  at  us  at  such 
times,  and  not  infrequently  he  wound  up  by  quoting  a  few  lines  of 
poetry  to  impress  on  us  the  importance  of  what  he  had  been  declaring. 
I  remember  distinctly  an  incident  that  occurred  when  he  was  at  work 
on  his  Universal  Algebra.  He  had  jumped  to  a  conclusion  which  he 
was  unable  to  prove  by  logical  deduction.  He  stated  this  fact  to  us 
in  the  lecture,  and  then  went  on,  "  GentleTiIEN  "  [here  he  raised  him- 
self on  his  toes],  "I  am  certain  that  my  conclusion  is  correct.  I  will 
WAGER  a  hundred  pounds  to  one  ,•  yes,  I  will  "WAGER  my  life  on  it." 
The  capitals  indicate  when  he  rose  on  his  toes  and  the  italics  when  he 
rocked  back  on  to  his  heels.  In  such  bursts  as  these  he  always  held  his 
hands  tightly  clenched  and  close  to  his  side,  while  his  elbows  stuck  out 
in  the  plane  of  his  body,  so  that  his  bended  arm  made  an  angle  of 
about  140O. 

"  Personally  I  had  considerable  contact  with  him,  as  I  did  work  under 
his  direction  that  made  it  necessary  for  me  to  see  him  at  his  rooms. 
On  such  occasions  I  always  made  an  engagement  with  him  two  or  three 
days  beforehand,  and  then  at  his  request  dropped  him  a  postal,  which 
reached  him  an  hour  or  two  before  I  went  and  reminded  him  that  I  was 
coming.  I  always  found  him  interested  in  my  work  and  full  of  sugges- 
tions. 

"  He  had  one  very  remarkable  peculiarity.  He  seldom  remembered 
theorems,  propositions,  etc.,  but  had  always  to  deduce  them  when  he 
wished  to  use  them.  In  this  he  was  the  very  antithesis  of  Oayley,  who 
was  thoroughly  conversant  with  everything  that  had  been  done  in  every 
branch  of  mathematics. 

"  I  remember  once  submitting  to  Sylvester  some  investigations  that 
I  had  been  engaged  on,  and  he  immediately  denied  my  first  statement, 
saying  that  such  a  proposition  had  never  been  heard  of,  let  alone 
proved.  To  his  astonishment,  I  showed  him  a  paper  of  his  own  in 
which  he  had  proved  the  proposition  ;  in  fact,  I  believe  the  object  of 
his  paper  had  been  the  very  proof  which  was  so  strange  to  him." 

By  request  of  Professor  Sylvester,  Professor  Cayley,  the  Sadlerian 
professor  of  pure  mathematics  in  Cambridge,  England,  was  associated 
in  the  mathematical  work  of  the  Johns  Hopkins  University  from  Jan- 
uary to  Jane,  1882.    Of  him  Dr.  Durfee  says :  "  His  subject  was  Abel- 


INFLUX  OF  FKENCH  MATHEMATICS.  269 

ian  and  Theta  Functions,  and  he  stack  closely  to  his  text.  While  his 
work  was  of  great  interest  and  iinporta,nce,  he  did  not  arouse  enthu- 
siasm and  diffuse  inspiration,  as  Sylvester  did." 

These  were  proud  days  for  the  Johns  Hopkins  TJniversity,  when  the 
two  greatest  living  English  mathematicians  were  lecturing  within  her 
walls. 

The  first  associate  appointed  in  mathematics,  when  the  university 
first  opened  in  1876,  was  Dr.  W,  B.  Story.  He  graduated  at  Harvard 
University  in  1871,  then  studied  for  some  years  in  Germany,  receiving 
the  degree  of  doctor  of  philosophy  at  the  University  of  Leipsio  in  1875. 
For  one  year  preceding  the  opening  of  the  Johns  Hopkins  University 
he  was  tutor  of  mathematics  at  Harvard.  At  the  Johns  Hopkins  Uni- 
versity his  lectures  and  his  original  researches  have  been  chiefly  in  geom- 
etry. He  was  for  several  years  associate  editor  of  the  American 
Journal  of  Mathematics. 

Dr.  Story  is  not  only  an  eminent  mathematician,  but  also  a  good 
teacher.  He  is  ever  ready  to  give  private  interviews  to  students  and 
to  explain  to  them  difficult  points,  or  offer  criticisms  and  suggestions 
upon  original  inquiries  which  the  student  may  be  engaged  in.  Dr.  Story 
is  an  admirable  lecturer,  clear,  logical,  deliberate,  proceeding  step  by 
step,  so  that  the  student  may  be  sure  to  follow  his  reasoning.  His  work 
on  the  blackboard  is  written  in  an  elegant  hand,  and  is  always  scrupu- 
lously accurate.  In  1884  the  university  secured  a  magnificent  set  of 
geometrical  models  for  the  study  of  surfaces.  Some  of  these  are  often 
brought  by  Dr.  Story  into  the  lecture-room  to  illustrate  his  subject.  In 
his  lectures  Dr.  Story  generally  follows  some  particular  text-book,  such 
as  Clebsch  on  Conic  Sections,  Salmon  on  Analytic  Geometry  of  Three 
Dimensions,  or  Steiner  on  Synthetic  Geometry,  but  he  often  brings  in 
researches  of  more  recent  date,  and  also  inquiries  of  his  own. 

Another  member  of  the  mathematical  staff  is  Dr.  Thomas  Craig.  He 
graduated  with  the  degree  of  civil  engineer  at  Lafayette  College  in  1875, 
was  one  of  the  first  persons  elected  to  a  fellowship  at  the  Johns  Hop- 
kins University,  and  in  1878  received  the  degree  of  doctor  of  philoso- 
phy. He  began  lecturing  at  the  university  when  he  was  a  student. 
After  graduation  he  was  connected  for  a  short  i^eriod  with  the  U.  S. 
Coast  and  Geodetic  Survey,  for  which  he  prepared  in  1879  a  Treatise 
on  the  Mathematical  Theory  of  Projections.  During  his  stay  in  Wash- 
ington he  studied  also  Theory  of  Functions  from  the  work  of  Konigs- 
berger,  under  the  direction  of  Professor  jSTewcomb,  of  the  Nautical  Al- 
manac. Dr.  Craig  has  made  the  theory  of  functions  and  differential 
equations  his  specialty.  He  has  not  only  kept  pace  with  the  most  re- 
cent rapid  advances  of  these  broad  and  deep  subjects,  but  has  added 
numerous  contributions  of  his  own.  Most  of  them  have  appeared  in  the 
American  Journal  of  Mathematics,  while  some  have  been  published  in 
foreign  journals.  He  is  working  on  subjects  which  are  receiving  ex- 
tensive development  in  the  hands  of  Fuchs,  Hermite,  Poincar6,  Appel, 


270  TEACHING   AND   HISTOKY    OF    MATHEMATICS. 

Darboux,  Picard,  and  others.  There  seem  to  be  altogether  too  few 
Americans  interested  in  this  line  of  work  and  prepared  to  participate 
in  its  advancement.  The  mind  of  Dr.  Oraig  moves  with  great  rapidity. 
A  quick  and  brilliant  student  finds  his  lectures  profitable  and  inspiring. 
Some  of  his  courses  on  differential  equations  and  the  theory  of  func- 
tions are  very  advanced  and  difl&cult,  and  can  be  followed  only  by  the 
maturest  of  students. 

Dr.  Craig  associates  with  the  students  familiarly.  It  has  been  his 
practice  to  invite  occasionally  students  to  his  house  to  spend  a  mathe- 
matical evening,  when  all  sorts  of  subjects  would  be  discussed  in  a  free 
and  easy  style. 

A  somewhat  more  recent  appointment  as  associate  in  mathematics  is 
that  of  Dr.  Fabian  Franklin.  He  graduated  at  the  Columbian  Univer- 
sity in  1869,  was  fellow  in  mathematics  from  1877  to  1879,  and  received 
the  degree  of  doctor  of  philosophy  in  1880.  He  was  appointed  assistant 
in  mathematics  before  taking  his  degree.  Franklin  always  took  great 
interest  in  Professor  Sylvester's  researches  while  the  latter  was  at  the 
Johns  Hopkins  University,  and  generally  was  at  work  on  similar  lines, 
while  Dr.  Story  and  Dr.  Craig  followed  more  generally  lines  of  inves- 
tigation of  their  own.  Some  of  the  articles  printed  in  the  American 
Journal  of  Mathematics  have  appeared  under  the  joint  authorship  of 
Sylvester  and  Franklin.  Professor  Sylvester  entertained  the  highest 
opinion  of  Dr.  Franklin. 

Dr.  Franklin  has  done  more  teaching  in  the  under-graduate  depart- 
ment than  the  other  members  of  the  mathematical  staff,  for  the  reason 
that  he  excels  them  all  in  his  power  of  imparting  instruction.  His 
teaching  power  is  indeed  great.  It  is  seldom  that  a  person  of  so  high 
mathematical  talent  is  as  good  an  instructor  of  younger  pupils.  Dr. 
Franklin  possesses  a  remarkably  quick  eye  for  short  methods.  The 
student  seldom  listens  to  one  of  his  lectures  in  which  proofs  are  not 
given  in  a  shorter,  simpler  manner  than  in  the  book  j  seldom  is  a  paper 
read  in  the  Mathematical  Society  which  is  not  followed,  in  the  ensuing 
discussion,  by  suggestions  by  Dr.  Franklin  of  a  shorter  method.  His 
papers  published  in  the  American  Journal  of  Mathematics  display  the 
same  power.  As  a  teacher  Dr.  Franklin  is  extremely  popular  among 
the  students. 

In  Dr.  Story,  Dr.  Craig,  and  Dr.  Franklin,  Professor  Sylvester  had 
an  eminently  efficient  corps  of  fellow-laborers.  Their  mathematical  re- 
searches have  made  their  names  favorably  known  wherever  advanced 
mathematics  finds  a  votary. 

The  instruction  for  graduates  during  the  time  that  Professor  Sylves- 
ter was  connected  with  the  university  was  as  follows  :  * 

Courses  of  Instruction,  Hours  per  Week,  and  Attendance,  1876-83. 

Determinants  and  Modern  Algebra:  Professor  Sylvester,  1876-77,  2d  half-year,  Shrs. 

(7);  1877-78,  2  lirs.  (5);  1878-79,  2  lira.  (8). 

•  Eleventli  Annual  Report  of  the  President  of  the  Johns  Hopkins  University,  p.  49. 


INFLUX  OF  FEENCH  MATHEMATICS.  271 

Theory  of  Numbers:  Prof  essor  Sylve  ster,  1879-80,  2  his.  (8) ;  1880-81,  Shra.  (6) ;  1881- 

82,  Ist  half-year,  2  hrs.  (7). 
Theory  of  Partitions :  Professor  Sylvester,  1882-83,  2d  half-year,  2  hra,  (10). 
Algebra  of  Multiple  Quantity  :  Professor  Sylvester,  1881-82,  2d.  half-year,  2  hrs.  (12); 

1883-84,  1st  half  year,  2  hrs.  (6). 
Theory  of  Substitutions :  Professor  Sylvester,  1882-83,  Ist  half-year,  2  hrs.  (9). 
Algebraical  Geometry  and  Abelian  and  Theta  Functions  :  Professor  Cayley,  1881-82, 

2d  half-year,  2  hrs.  (14). 
Quaternions:  Dr.  Story,  1877-78,  2  hrs.  (2);  1879-80,  3  hrs.  (4);  1881-82,  3  hrs.  (7); 

1882-83,  2d  half-year,  3  hrs.  (4). 
Higher  Plane  Curves:  Dr.  Story,  1880-81,  2  hrs.  (5) ;  1881-82,  Ist  half  year,  3  hrs.  (1) ; 

1883-84,  2  hrs.  (2). 
Solid  Analytic  Geometry  (General  Theory  of  Surfacea  and  Curves) :  Dr.  Story,  1881- 

82,  2d  half-year,  3  hrs.  (1) ;  1882-83,  1st  half-year,  3  hrs.  (6). 
Theory  of  Geometrical  Congruences :  Dr.  Story,  1882-83,  2d  half-year,  2  hra.  (4). 
Modern  Synthetic  Geometry :  Dr.  Franklin,  1877-78,  2  hra.  (2). 
Theory  of  Invariants :  Dr.  Story,  1882-83,  10  lectures  (8) ;  1883-84,  3  hrs.  (6). 
Determinants  :  Dr.  Franklin,  1880-81,  1st  half-year,  2  hrs.  (9) ;  1882-83,  20  lectures 

(9). 
Modem  Algebra :  Dr.  Franklin,  1880-81,  2d  half-year,  2  hrs.  (6) ;  1881-82,  2d  half- 
year.  2  hrs.  (6). 
Elliptic  Functions :  Dr.  Story,  1878-79,  2  hrs.  (2);  1879-80  (continuation  of  the  pre- 
vious year's  course),  3  hrs.  (4)  ;  Dr.  Craig,  1881-82,  3  hrs.  (8) ;  1883-84,  3  hrs.  (4). 
Elliptic  and  Theta  Functions :  Dr.  Craig,  1882-83,  3  hrs.  (10) ;  1883-84,  3  hrs.  (2). 
General  Theory  of  Functions,  including  Riemann's  Theory :  Dr.  Craig,  1879-80,   30 

lectures  (2) ;  1880-81,  Ist  half-year,  3  hrs.  (3). 
Spherical  Harmonics :  Dr.   Craig,  1878-79,  10  lectures  (6) ;  1879-80,  20  lectures  (6) ; 

1881-82,  Ist  half-year,  2  hrs.  (4) ;  1883-84,  2d  half-year,  1  hr.  (4). 
Cylindric  or  Bessel's  Functions:  Dr.  Craig,  1879-80,  10  lectures  (2). 
Partial  Differential  Equations :  Dr.  Craig,  1880-81,  2d  half-year,  2  hrs.  (5) ;  1881-82, 

2d  half-year,  3  hrs.  (9) ;  1882-83,  2d  half-year,  2  hrs.  (2) ;  1883-84,  2d  half-year,  2 

hrs.  (4). 
Calculus  of  Variations :  Dr.  Craig,  1879-80,  12  lectures  (9) ;  1881-82, 1st  half-year,  2 

hrs.  (8) ;  1882-83,  1st  half-year,  2  hrs.  (6). 
Definite  Integrals :  Dr.  Craig,  1876-77,  Ist  half-year,  3  hrs.  (5) ;  1882-83, 1st  half-year, 

2  hrs.  (2). 

Mathematical  Astronomy:  Dr.  Story,  1877-78,  3 hrs.  (2);  1882-83,  3  hrs,  (2);  1883-84, 

3  hrs.  (2). 

Elementary  Mechanics :  Dr..  Craig,  1876-77,  2d  half-year  (8). 

Statics:  Dr.  Franklin,  1882-83,  2d  half-year,  3  hrs.  (5). 

Analytic  Mechanics :  Dr.  Craig,  1877-78,  Ist  half-year  (6) ;  Dr.  Story,  1880-81,  2d 

half-year,  2  hrs.  (6) ;  Dr.  Craig,  1881-82,  Ist  half-year,  3  hrs.  (8) ;  1882-83,  Ist 

half-year,  3  hrs.  (4) ;  Dr.  Franklin,  1883-84,  3  hrs  (6). 
Theoretical  Dynamics :  Dr.  Craig,  1878-79, 15  lectures  (6)  ;  1883-84,  2  hra  (5). 
Mathematical  Theory  of  Elasticity :  Dr.  Story,  1876-77,  2d  half-year,  2  hrs.  (4) ; 

1877-78, 2  hrs.  (2) ;  Dr.  Craig,  1881-82,  3  hrs.  (4) ;  1883-84,  2d  haK-year,  2  hrs.  (3). 
Hydrodynamics:  Dr.  Craig,  1878-79,  24  lectures  (7);  1880-81,  1st  half-year,  2  hra. 

(6) ;  2d  half-year,  4  hrs.  (3)  ;  1882-83,  2d  half-year,  3  hrs.  (5). 
Mathematical  Theory  of  Sound :  Dr.  Craig,  1883-84,  3  hrs.  (5). 

It  may  be  of  interest  to  give  a  list  of  the  advanced  students  of  math- 
ematics  during  the  seven  years  that  Sylvester  was  connected  with  the 
Johns  Hopkins  University,  and  their  present  occupation.  Dr.  Craig 
and  Dr.  Franklin  are,  as  we  have  seen,  instructors  in  mathematics  at 
the  Johns  Hopkins.    The  list  continues  as  follows :    G.  B.  Halsted, 


272  TEACHING   AND    HTSTOEY    OF   MATHEMATICS. 

professor  of  mathematics,  University  of  Texas ;  W.  I.  Stringham,  pro- 
fessor of  matliematics,  University  of  California ;  0.  A.  Van  Yelzer,  pro- 
fessor of  mathematics,  University  of  Wisconsin;  O.  H.  Mitchell,  pro- 
fessor of  mathematics,  Marietta  College,  Ohio  ;  E.  W.  Prentiss,  in  the 
office  of  the  U.  S.  Nautical  Almanac,  Washington ;  H.  M.  Perry,  in- 
structor in  mathematics,  Cascadilla  School,  Ithaca,  E.  T. ;  W.  P.  Durfee, 
professor  of  mathematics,  Robart  College,  N.  Y. ;  G.  S.  Ely,  examiner, 
U,  S.  Patent  Office ;  E.  W.  Davis,  professor  of  mathematics,  University 
of  South  Carolina;  A.  S.  Hathaway,  instructor  in  mathematics,  Cornell 
University ;  G.  Bissing,  examiner,  U.  S.  Patent  Office ;  A.  L.  Daniels, 
instructor  in  mathematics,  Princeton  College,  1883-84. 

The  success  in  training  students  for  independent  research  has  been 
very  great.  To  convince  himself  of  this,  the  reader  need  only  ex- 
amine the  abstracts  of  papers  prepared  by  students,  which  have  been 
published  in  the  Johns  Hopkins  Circulars  and  in  the  American  Journal 
of  Mathematics.  Each  one  of  the  names  given  above  will  be  found  to 
appear  repeatedly  in  those  publications,  as  a  contributor. 

In  December,  1883,  Professor  Sylvester  started  for  England  to  enter 
upon  his  new  duties  as  Savilian  professor  of  geometry  in  the  University 
of  Oxford.  The  robe  of  the  departing  prophet  dropped  upon  the  shoul- 
ders of  Professor  H"ewcomb.  No  American  would  have  been  more 
worthy  of  succeeding  Sylvester.  As  an  astronomer  his  name  has  long 
shone  with  a  luster  Tvhich  fills  with  pride  every  American  breast. 

Simon  Newcomb  was  born  in  Wallace,  Nova  Scotia,  in  1835.  After 
being  educated  by  his  father  he  engaged  for  some  time  in  teaching.  He 
came  to  the  United  States  at  the  age  of  eighteen,  and  was  engaged  for  two 
years  as  teacher  in  Blaryland.  Th'ere  he  became  acquainted  with 
Joseph  Henry,  of  the  Smithsonian  Institution,  and  Julius  E.  Hilgard,  of 
the  U.  S.  Coast  Survey.  Eecognizing  his  talent  for  mathematics,  they 
secured  for  him,  in  1857,  a  position  as  computer  on  the  Nautical  Alma- 
nac, which  was  then  published  in  Cambridge,  Mass.  In  Cambridge  he 
came  under  the  influence  of  Prof.  Benjamin  Peirce.  In  the  catalogues 
of  1856  and  1857  his  name  appears  as  a  student  of  mathematics  in  the 
Sheffield  Scientific  School.  He  graduated  in  1858,  and  continued  as  a 
graduate  student  for  three  years  thereafter.  He  was  then  appointed 
professor  of  mathematics  in  the  U.  S.  Navy,  and  stationed  at  the  Naval 
Observatory.  He  was  chief  director  of  a  commission  created  by  Con- 
gress to  observe  the  transit  of  Venus  in  1874.  In  that  year  the  Royal 
Society  of  England  a,warded  him  a  gold  medal  for  his  Tables  of  Uranus 
and  Neptune.  In  1870  lie  undertook  to  investigate  the  errors  of  Han- 
sen's Lunar  Tables  as  compared  with  observations  prior  to  1850.  The 
results  of  this  onerous  task  were  published  in  1878.  In  the  years  1880 
to  1882  he  and  Michelsou  measured  the  velocity  of  light  by  operations 
on  such  a  large  scale  and  such  refined  methods  as  to  throw  in  the  shade 
all  earlier  efibrts  of  a  similar  kind.  Eor  the  purpose  of  this  measure- 
ment they  set  up  fixed  and  revolving  mirrors  on  opposite  shores  of  the 
Potomac,  at  a  distance  of  nearly  4  kilometers. 


INFLUX  OP  FEENCH  MATHEMATICS.  273 

Since  1877  he  has  been  in  charge  of  the  office  of  the  American  Ephem- 
eris  and  Nautical  Almanac.  Since  18G7  that  office  has  been  in  Wash- 
ington, instead  of  Cambridge.  Professor  Kewcomb's  predecessor  in  this 
office  was  J.  H.  G.  Coffin,  who  in  1877  was  placetl  on  the  retired  list, 
having  been  senior  professor  of  mathematics  in  the  Navy  since  1848. 

Professor  Newcomb  has  quite  a  large  corps  of  assistants  in  Wash- 
ington. His  researches  in  astronomy  during  the  last  ten  or  twelve 
years  have  been  described  in  the  Nation  of  September  6,  1888,  as  fol- 
lows: 

"  The  general  object  of  this  work  is  the  determination  of  the  form, 
size,  and  position  of  the  orbits  of  all  the  large  planets  of  the  solar  sys- 
tem, from  the  best  and  most  recent  observations,  and  the  preparation 
of  entirely  new  and  uniform  tables  for  predicting  the  future  positions 
of  these  objects.  The  first  of  the  four  sections  of  the  work  relates  to  the 
general  perturbations  of  the  planets  by  each  other,  and  the  part  already 
in  hand  comprises  the  four  inner  planets,  Mercury,  Venus,  the  Earth, 
and  Mars,  in  which  fourteen  pairs  of  planets  come  into  play.  Twelve 
of  these  were  completed  some  months  since,  and  only  the  action  of  Ju- 
piter on  Venus  and  Mars  remained  undetermined.  In  the  next  place, 
the  older  observations  of  the  planets  must  be  recalculated,  and  thus 
problems  constantly  arise  which  can  not  be  met  by  general  rules.  All 
the  observations  at  Greenwich  from  1765  to  1811  have  been  completely 
reduced  with  modern  data.  Earlier  Greenwich  observations  were  sim- 
ilarly treated  by  Dr.  Auwers,  of  Berlin,  who  liberally  presented  the  com- 
plete calculations  as  his  contribution  to  the  work  of  the  Nautical 
Almanac  Office.  In  a  recent  report  of  this  work  Professor  Newcomb 
gives  further  details  of  his  progress  in  the  treatment  of  other  classes 
of  planetary  observations. 

"  Following  this  collation  of  all  the  available  observations  of  each 
planet,  comes  the  theoretical  preparation  of  their  corresponding  posi- 
tions at  the  time  of  observation.  This  forms  the  most  laborious  and 
difficult  part  of  the  work ;  and  had  Leverrier's  tables,  the  best  hitherto, 
been  used  without  modification.  Professor  Newcomb  would  have  found 
it  impractiable  to  complete  it  with  the  number  of  computers  at  his  com- 
mand ;  but  he  has  skillfully  avoided  the  difficulty  by  a  reconstruction 
of  Leverrier's  work  in  such  form  that  it  should  be  much  less  laborious 
to  use,  while  sufficiently  accurate  for  the  purpose  required.  These  theo- 
retical positions  must  next  be  compared  directly  with  the  observations, 
one  by  one,  and  the  dififerences  between  the  two  are  then,  by  suitable 
mathematical  processes,  constrned  as  implying  the  nature  and  amount 
of  certain  corrections  to  the  planet's  motion  in  its  orbit.  More  than  a 
full  year  must  still  elapse,  says  Professor  Newcomb,  before  the  work  ou 
the  four  inner  planets  will  have  advanced  to  the  stage  where  this  direct 
comparison  is  ready  to  be  made.  There  remain  the  four  outer  planets, 
on  the  two  more  important  of  which,  Jupiter  and  Saturn,  Mr.  Hill,  of 
the  same  office,  has  been  engaged  for  many  years,  and  his  new  theory 
881~No.  3—18 


274  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

of  their  complicated  motion  is  already  in  the  printer's  hands.  The  two 
outer  planets,  Uranus  and  Neptune,  have  not  yet  been  begun." 

Of  Professor  Newcomb's  labors  Professor  Cay  ley  has  said :  "  Professor 
iTewcomb's  writings  exhibit,  all  of  them,  a  combination  on  the  one  hand 
of  mathematical  skill  and  power,  and  on  the  other  of  good  hard  work, 
devoted  to  the  furtherance  of  astronomical  science." 

His  book  on  Popular  Astronomy  (1877)  is  well  known.  It  has  been 
republished  in  England  and  translated  into  German.  The  treatise  on 
Astronomy  by  Newcomb  and  Holden,  and  their  "  Shorter  Course"  on 
Astronomy,  are  works  which  have  been  introduced  as  text-books  into 
our  colleges  almost  universally. 

Professor  Newcomb's  scientific  work  has  not  been  confined  to  astron- 
omy. He  has  carried  on  investigations  on  subjects  purely  mathemati- 
cal. One  of  the  most  important  is  his  article  on  ''Elementary  Theo- 
rems Eelating  to  the  Geometry  of  a  Space  of  Three  Dimensions  and  of 
Uniform  Positive  Curvature  in  the  Fourth  Dimension,"  published  in 
Borchardt's  Journal,  Bd.  83,  Berlin,  1877.  Full  extracts  of  this  very 
important  contribution  to  non-Euclidian  geometry  are  given  in  the 
Enclyclopsedia  Britannica,  article  "  Measurement."  It  is  gratifying  to 
know  that  through  Professor  Newcomb  America  has  done  something 
toward  developing  the  far-reaching  generalizations  of  non-Euclidian 
geometry  and  hyper-space.  In  Volume  I  of  the  American  Journal  of 
Mathematics  he  has  a  note  "  On  a  Class  of  Transformations  which  Sur- 
faces may  Undergo  in  Space  of  more  than  Three  Dimensions,"  in  which 
he  shows,  for  instance,  that  if  a  fourth  dimension  were  added  to  space, 
a  closed  material  surface  (or  shell)  could  be  turned  inside  out  by  sim- 
ple flexure  without  either  stretching  or  tearing.  Later  articles  have 
been  on  the  theory  of  errors  in  observations.  In  former  years  he  also 
contributed  to  the  Mathematical  Monthly  and  the  Analyst. 

Professor  Newcomb  has  written  a  series  of  college  text-books  on 
mathematics.  In  1881  appeared  his  Algebra  for  Colleges  and  his  Ele- 
ments of  Geometry ;  in  1882  his  Trigonometry  and  Logarithms,  and 
School  Algebra ;  in  1884  his  Analytical  Geometry  and  Essentials  of 
Trigonometry ;  in  1887  his  Differential  and  Integral  Calculus.  These 
works  have  been  favorably  reviewed  by  the  press,  and  are  everywhere 
highly  respected.  Professor  l»fewcomb's  fundamental  idea  has  been  to 
lead  up  to  new  and  strange  conceptions  by  slow  and  gradual  steps. 
"AU  mathematical  conceptions  require  time  to  become  engrafted  upon 
the  mind,  and  the  more  time  the  greater  their  abstruseness."  The  stu- 
dent is  gradually  made  familiar  in  these  books  with  the  conceptions  of 
variables,  functions,  increments,  infinitesimals,  and  limits,  long  before 
he  takes  up  the  calculus,  so  in  the  study  of  the  calculus  he  is  not  con- 
fronted, at  the  outset  and  all  at  once,  by  a  host  of  new  and  strange 
ideas,  but  possesses  already  a  considerable  degree  of  familiarity  with 
them.  With  the  publication  of  E"ewcomb's  Algebra  has  begun  a  con- 
siderable "shaking"  of  the  "  dry  bones"  in  this  science,  and  we  now 
possess  works  on  this  subject  that  are  of  considerable  merit, 


INFLUX    OF    FRENCH   MATHEMATICS.  275 

Professor  Newcomb  stadias  political  econoTjiy  as  a  recreation,  and 
every  now  and  then  ther©  is  a  commotion  in  the  camp  of  political  econ- 
omists, caused  by  a  bomb  thrown  into  their  midst  by  Professor  New- 
comb,  in  the  form  of  some  Fjia.gazine  article  or  book. 

In  1884  Professor  Newcomb  added  to  his  duties  as  superintendent  of 
the  Nautical  Almanac  that  of  professor  of  mathematics  and  astronomy 
at  the  Johns  Hopkins  Uni¥ersity.  He  generally  delivers  at  that  in- 
stitution two  lectures  per  week.  The  eStect  of  his  connection  with 
the  mathematical  department  has  been  that  the  mathematical  course 
is  more  thoroughly  systeaiatized  and  more  carefully  graded  than 
formerly,  and  that  the  attention  of  students  is  drawn  also  to  higher 
astronomy,  theoretical  and  practical.  An  observatory  for  instruction 
is  now  provided  by  the  university.  Besides  a  telescope  of  9^  inches 
aperture  there  is  a  meridian  circle  with  collimators,  mercury-basin, 
and  other  appliances.  Professor  Newcomb  entered  upon  his  duties 
at  the  Johns  Hopkins  University  in  1884  by  giving  a  course  of  lect- 
ures on  celestial  mechanics.  Among  other  things  it  embraced  his 
own  paper  on  the  "  Development  of  the  Perturbative  Function  and 
its  Derivative  in  Sines  and  Co-sinesi  of  the  Eccentric  Anomaly  and  in 
Powers  of  the  Eccentricities  and  Inclinations."  The  lectures  were  well 
attended  by  the  graduate  students.  At  the  blackboard  Professor 
Newcomb  does  not  manipulate  the  crayon  with  sogreat  dexterity  as  do 
his  associates,  who  have  been  in  the  lecture-room  all  their  lives,  but  his 
lectures  are  clear,  instructive,  original,  and  popular  among  the  students. 

Since  the  departure  of  Professor  Sylvester  the  following  courses  of 
lectures  have  been  given  to  graduate  students: 

Courses  of  Instruction,  Hours  per  Week,  and  Attendance,  1884-'88.* 

Analytical  and  Celestial  Mechanics :  Prof.  Newcomb,  1884-'85,  2  lira.  (11). 
Practical  and  Theoretical  Astronomy  :  Prof.  Newcomb,  1885-'86,  2  hrs.  (9) ;  1886-'87, 

2  hrs.  (7). 
Theory  of  Special  Perturbations :  Prof.  Newcomb,  1887-'88,  1st  half-year,  2  hra. 
History  of  Astronomy :  Prof.  Newcomb,  1887-88,  March  and  April,  2  hrs. 
Computation  of  Orbits :  Prof.  Newcomb,  1887-88,  May,  2  hrs. 
Theory  of  Numbers:  Dr.  Story,  1884-'85,  1st  half-year,  2  hrs.  (9). 
Modern  Synthetic  Geometry:  Dr.  Story,  1884-85,  Ist  half-year,  3  hra.  (8). 
Introductory  Course  for  Graduates:  Dr.  Story,  1884-'85,  5 hrs.  (10) ;  1885-'86,  5 hrs. 

(7) ;  1886-'87,  5  hrs.  (10) ;  1887-'88,  5  hrs. 
Modern  Algebra :  Dr.  Story,  1884-'85,  2d  half-year,  2  hrs.  (9). 
Quaternions :   Dr.  Story,   1884-'85,  2d  half-year,  3  hrs.   (8) ;    1886-'87.  3  hrs.  (5) ; 

1887-'88,  3  hrs. 
Finite  Differences  and  Interpolation  :  Dr.  Story,  1885-86,  1st  half-year,  2  hrs.  (5). 
Advanced  Analytic  Geometry :  Dr.  Story,  1885-'86,  3  hrs.  (4);  1886-'87,  2  hrs.  (8)j 

1887-'88,  2  hrs. 
Theory  of  Probabilities :  Dr.  Story,  1885-'86,  2d  half-year,  2  hrs.  (5). 
Calculus  of  Variations :  Dr.  Craig,  1884-'85,  1st  half-year,  2  hrs.  (5). 
Theory  of  Functions  :  Dr.  Craig,  1884-'85,  3  hrs.  (5)  ;  1885-'86,  1st  half-year,  3  hra. 

(4) ;  188G-'87,  3  hrs.  (6) ;  1887-'88,  Ist  half- j  ear,  3  hrs. 
Hydrodynamics :  Dr.  Craig,  1884-'85,  1st  half-year,  3  hrs.  (6) ;  1885-'83, 1st  half-year, 

3  hrs.  (4)  ;  1886-'87, 1st  half-year,  3  hra.  (4) ;  1887-'88,  1st  half-year,  3  hrs. 


276  TEACHINO   AND    HISTOEY    OF   MATHEMATICS. 

Linear  Differential  Equations:  Dr.  Craig,  1884-'85,  2d  half-year,  3  hrs.  (3);  1885-'86, 

2  hrs.  (4) ;  1887-'88,  2d  half-year,  2  hrs. 
Theoretical  Dynamics :  Dr.  Craig,  1887-'88,  2d  half-year,  2  hrs. 
Differential  Equations:  Dr.  Craig,  1887-'88,  2  hrs. 

Mathematical  Theory  of  Elasticity :  Dr.  Craig,  1835-'86,  2d  half-year,  3  hrs.  (4). 
Elliptic  and  Abelian  Functions  :  Dr.  Craig,  1885-'86,  2d  half-year,  3  hrs.  (4) ;  1885-'87, 

1st  half-year,  2  hrs.  (6). 
Abelian  Functions  :  Dr.  Craig,  1887-'88,  2  hrs. 
Problems  in  Mechanics :  Dr.  Franklin,  1884-'85,  2  hrs.  (5) ;  1885-'86,  2  hrs.  (6) ;  1886- 

'87,  2  hrs.  (8) ;  1887-'88,  2  hrs. 

Since  the  fall  of  1884  Dr.  Story  has  been  giving  every  year  an  Intro- 
ductory Course  to  graduate  students,  which  consists  of  short  courses  of 
lectures  on  the  leading  branches  of  higher  mathematics.  They  are  in- 
tended to  give  the  student  a  general  view  of  the  whole  field,  which 
afterward  he  is  to  enter  upon  and  study  in  its  details. 

The  Johns  Hopkins  University  went  into  operation  primarily  as  a 
University,  giving  instruction  to  students  who  had  graduated  from  col- 
lege. A  regular  college  course  was,  however,  organized,  and  it  has  been 
growing  rapidly  from  year  to  year.  In  the  college  the  student  has  the 
choice  between  several  parallel  curricula,  which  are  assumed  to  be  equally 
honorable,  liberal,  and  difficult,  and  which  therefore  lead  to  the  same 
degree  of  bachelor  of  arts.  Seven  groups  have  been  arranged.  Some 
of  them  embrace  no  mathematics  at  all ;  but,  in  those  courses  where  it  does 
enter,  the  instruction  is  very  thorough.  Take,  for  instance,  Dr.  Story's 
lectures  on  conic  sections;  the  method  of  treatment  is  entirely  modern, 
and  presupposes  a  knowledge  of  determinants.  A  syllabus  has  been  pre- 
pared for  the  use  of  the  students.  The  lectures  resemble  the  course 
given  in  the  work  of  Clebsch.  The  student  who  may  have  studied 
such  books  as  Loomis's  Analytical  Geometry,  and  who  may  labor  with 
the  conceit  that  he  has  mastered  analytical  geometry  and  conic  sections, 
will  soon  discover  that  he  has  learned  only  the  ABC,  and  that  he  is 
wholly  ignorant  of  the  more  elegant  methods  of  modern  times. 

Connected  with  the  mathematical  department  of  the  university  has 
always  been  a  mathematical  seminary,  which  during  the  time  of  Syl- 
vester constituted  in  fact  the  mathematical  society  of  the  university. 
The  meetings  were  held  monthly.  In  it  the  instructors  and  more  ad- 
vanced students  would  present  and  discuss  their  original  researches. 
Care  was  taken  to  eliminate  papers  of  little  or  no  value  by  immature 
students.  Professor  Sylvester  generally  presided.  "  If  you  were  fortu- 
nate," says  Dr.  B.  W.  Davis,  "  you  had  your  paper  first  on  the  pro- 
gram. Short  it  must  be  and  to  the  point.  Sylvester  would  be  pleased. 
Then  came  his  paper,  or  two  of  them.  After  him  came  the  rest,  but  no 
show  did  theystand ;  Sylvester  was  dreaming  of  his  own  higher  flights 
and  where  they  would  yet  carry  him." 

Since  the  time  of  Fewcomb  this  mathematical  seminary  has  been 
called  the  Mathematical  Society.  It  is  carried  on  in  the  same  way  as 
before.    Three  mathematical  seminaries  proper  have  since  existed,  one 


MATHEMATICAL  JOUENALS.  277 

conducted  by  Professor  Newcomb,  another  by  Dr.  Story,  and  the  third 
by  Dr.  Oraig.  The  meetings  are  held  in  the  evening,  and  weekly.  Each 
instructor  selects  for  his  seminary  topics  from  his  special  studies ;  Is&w- 
comb,  astronomical  subjects;  Story,  geometrical  subjects  or  quater- 
nions j  Craig,  theory  of  functions  or  differential  equations.  Professor 
Newcomb's  seminary  work  is  closely  connected  with  his  lectures.  The 
student  elaborates  some  particular  points  of  the  lectures  or  makes  prac- 
tical application  of  the  principles  involved.  In  one  case  the  compu- 
tation of  the  orbit  of  a  comet  was  taken  up.  Dr.  Story,  in  the  year 
1885-86,  took  up  the  subject  of  plane  curves  for  his  seminary,  and  dwelt 
considerably  on  quartic  and  quintic  curves,  giving  matter  from  Mobius 
and  Zeuthen,  and  the  result  of  his  own  study  on  quintics.  The  stu- 
dent was  expected,  if  possible,  to  begin  where  he  had  left  off  and  carry 
on  investigations  along  lines  pointed  out  by  him.  Dr.  Story's  talk  on 
this  subject  in  this  seminary  suggested  to  one  of  the  students  a  subject 
of  a  thesis  for  the  doctor's  degree.  In  the  fall  of  1888  Dr.  Story  began 
his  seminary  work  with  the  seventeenth  example,  p.  103,  in  Tait's  Qua- 
ternions. Dr.  Craig's  seminary  has  generally  been  upon  subjects  in  con- 
tinuation and  extension  of  those  upon  which  he  is  lecturing  at  the  time. 
If,  for  instance,  he  is  lecturing  on  functions,  following  the  "  Cours  de 
M.  Hermite,^^  he  may  in  his  seminary  bring  up  matter  from  Briot  and 
Bouquet.  At  other  times  he  has  introduced  work  into  his  seminary 
intended  to  be  preparatory  to  certain  advanced  courses  which  he 
expected  to  offer. 

Mathematical  Journals. 

The  mathematical  journals  which  we  are  about  to  discuss  were  of  a 
much  higher  grade  than  those  of  preceding  years.  First  in  order  of 
time  is  the  Mathematical  Miscellany,  a  semi-annual  publication,  edited 
by  Charles  Gill.  He  was  teacher  of  mathematics  at  the  St.  Paul's  Col- 
legiate Institute  at  Flushing,  Long  Island.  Eight  numbers  were  pub- 
lished ;  the  first  in  .February,  1836,  and  the  last  in  November,  1839. 
Like  many  other  journals  of  this  kind,  it  had  a  Junior  and  Senior  de- 
partment— the  former  for  young  students,  the  latter  for  those  more 
advanced.  The  first  number  was  entirely  the  work  of  the  editor,  ex- 
cepting two  or  three  new  problems.  Mr.  Gill  was  much  interested  in 
Diophantine  analysis.  In  1848  he  published  a  little  book  on  the  Ap- 
plication of  Angular  Analysis  to  the  Solution  of  Indeterminate  Prob- 
lems of  the  Second  Degree,  which  contains  some  of  his  investigations 
on  this  subject. 

Another  enthusiastic  worker  in  the  field  of  Diophantine  analysis,  and 
a  frequent  contributor  to  Gill's  journal,  was  William  Lenhart,  a  favorite 
pupil  of  Robert  Adrain.  Having  been  afEicted  for  twenty-eight  years 
with  a  spasmodic  affection  of  the  limbs,  occasioned  by  a  fall  in  early 
life,  which  confined  him  in  a  measure  to  his  room,  he  had  devoted  a 
considerable  portion  of  his  time  to  Diophantine  analysis.    To  him  is 


'278  TEACHING   AND  SISTORY   OP   MATHEMATICS. 

attributed  the  solution  of  the  problem,  to  divide  unity  into  six  parts 
such  that,  if  unity  be  added  to  each,  the  sums  will  be  cubes. 

The  evident  defect  in  Lenhart's  processes  was  their  tentative  char- 
acter. In  fact,  this  criticism  applies  to  all  work  done  in  Diophantine 
analysis  by  American  computers,  down  to  the  present  time.  It  is  true 
even  of  old  Diophantus  himself.  To  this  ancient  Alexandrian  alge- 
braist, who  is  the  author  of  the  earliest  treatise  on  algebra  extant,  as 
well  as  to  his  American  followers  of  modern  times,  general  methods 
were  quite  unknown.  Each  problem  has  its  own  distinct  method,  which 
is  often  useless  for  the  most  closely  related  problems.  It  has  been  re- 
marked by  H.  Hankel  that,  after  having  studied  one  hundred  solu- 
tions of  Diophantus,  it  is  di£6.cult  to  solve  the  one  hundred  and  first. 
It  is  to  be  regretted  that  American  students  should  have  wasted  so 
much  time  over  Diophantine  analysis,  instead  of  falling  in  line  with 
European  workers  in  the  theory  of  numbers  as  developed  by  Gauss  and 
others.  Previous  to  the  publication  of  the  American  Journal  of  Math- 
ematics, our  journals  contained  no  contributions  whatever  on  the  theory 
of  numbers,  excepting  the  Mathematical  Miscellany,  which  had  some 
few  articles  by  Benjamin  Peirce  and  Theodore  Strong,  which  involved 
Gaussian  methods.  Among  the  contributors  to  the  Mathematical  Mis- 
cellany were  Theodore  Strong,  Benjamin  Peirce,  Charles  Avery,  Mar- 
cus Catlin  of  Hamilton  College,  George  E.  Perkins,  O.  Eoot,  William 
Lenhart,  Lyman  Abbott,  jr.,  B.  Docharty,  and  others. 

The  next  mathematical  periodical  was  the  Cambridge  Miscellany  of 
MathematicSj  Physics,  and  Astronomy,  edited  by  Benjamin  Peirce  and 
Joseph  Lovering,  of  Harvard,  and  published  quarterly.  The  last  prob- 
lems proposed  in  Gill's  journal  were  solved  here.  Four  numbers  only 
were  published,  the  first  in  1842.  The  list  of  contributors  to  this  jour- 
nal was  about  the  same  as  to  the  preceding.  The  most  valuable  arti- 
cles were  those  written  by  the  editors. 

During  the  next  fifteen  years  America  was  without  a  mathematical 
journal  5  but  in  1858,  J.  D.  Eunkle,  of  the  Nautical  Almanac  office  in 
Boston,  started  the  Mathematical  Monthly.  He  has  since  held  the  dis- 
tinguished position  of  professor  of  mathematics  at  (and,  for  a  time, 
president  of)  the  Massachusetts  Institute  of  Technology,  where  he  has 
been  especially  interested  in  developing  the  department  of  manual 
training.  As  will  be  seen  presently,  the  time  for  beginning  the  publi- 
cation of  a  long-lived  mathematical  journal  was  not  opportune.  Three 
volumes  only  appeared.  On  a  fly-leaf  the  editor  announced  to  his  sub- 
scribers that  over  one-third  of  the  subscribers  to  Volume  I  discontin- 
ued their  subscriptions  at  the  close.  "  I  have  supposed,"  he  says, 
"  that  those  who  continued  their  subscription  to  the  second  volume  would 
not  be  so  likely  to  discontinue  it  to  the  third  volume,  and  I  have  made 
my  arrangements  accordingly.  If,  however,  any  considerable  number 
should  discontinue  now,  it  will  be  subject  to  very  serious  loss.  *  *  * 
I  ask  as  a  favor  for  all  to  continue  to  Volume  III,  and  notify  me  during 


MATHEMATICAL  JOURNALS.  279 

the  year  if  they  intend  to  discontinue  at  its  close.  I  shall  then  know 
whether  to  begin  the  fourth  volume.  I  shall  not  realize  a  dollar."  This 
announcement  discloses  obstacles  which  all  our  journals  that  have  been 
dependent  entirely  upon  their  subscribers  for  financial  support  have 
had  to  encounter,  and  which  none  except  the  more  recent  could  long 
resist.  Moreover,  the  Civil  War  was  now  at  hand.  "  On  account  of 
the  present  disturbed  state  of  public  affairs,  the  publication  of  the 
Mathematical  Monthly  will  be  discontinued  until  further  notice."  This 
was  the  end  of  the  Monthly,  in  1861. 

The  salient  features  in  the  plan  upon  which  the  periodical  was  con- 
ducted, as  stated  by  David  S.  Hart,*  were:  "The  publication  of  five 
problems  in  each  number,  adapted  to  the  capacities  of  the  young  stu- 
dents, to  be  answered  in  the  third  succeeding  number.  The  insertion 
of  notes  and  queries,  short  discussions  and  articles  of  a  fragmentary 
character,  too  valuable  to  be  lost;  and,  lastly,  essays  not  exceeding 
eight  pages,  on  various  subjects,  in  all  departments  of  mathematics. 
Besides,  there  were  notices  and  reviews  of  the  mathematical  works 
issued,  both  old  and  new.  Among  the  most  interesting  articles  are  the 
account  of  the  comet  of  Donati,  with  elegant  descriptive  plates,  written 
by  the  astronomical  professor  of  Harvard  University  (Vol.  I,  Kos.  2  and 
3)  J  a  complete  catalogue  of  the  writings  of  John  Herschel  (Vol.  Ill, 
Ko.  7) ;  articles  on  indeterminate  analysis,  by  Eev.  A.  D.  Wheeler,  of 
Brunswick,  Me.  (Vol.  II,  Nos.  1,  6,  and  12),  and  the  Diophantine  analy- 
sis (Vol.  Ill,  ¥o.  11).  Other  articles  on  the  Diophantine  analysis  by 
Mr.  Wheeler  would  have  been  inserted,  if  the  Mathematical  Monthly 
had  been  continued.  *  The  Economy  and  Symmetry  of  the  Honey-bees^ 
Cells,'  by  Chauncey  Wright  (Vol.  II,  No.  9).  Simon  Newcomb  gives  sev- 
eral interesting  '  Notes  on  Probabilities.'  In  Vol.  II,  No.  2,  there  is  an 
article  containing  a  complete  list  of  the  writings  of  Nathaniel  Bowditch, 
accompanied  with  short  sketches  of  the  same,  which  is  extremely  inter- 
esting. *  *  *  The  periodical  is  embellished  by  portraits  of  N.  Bow- 
ditch,  Prof.  Benjamin  Peirce,  and  Sir  John  Herschel,  which  are  finely 
executed."  The  Monthly  presented  a  very  neat  appearance  to  the  eye. 
In  the  mathematical  notation  employed  and  in  the  treatment  of  mathe- 
matical subjects,  Benjamin  Peirce's  influence  was  clearly  perceptible. 
From  a  scientific  point  of  view,  the  Monthly  excelled  any  of  its  prede- 
cessors. 

Since  1861,  we  had  no  mathematical  periodical  in  the  United  States 
for  thirteen  years.  In  January,  1874,  was  published  in  Des  Moines, 
Iowa,  The  Analyst:  A  Monthly  Journal  of  Pure  and  Applied  Mathemat- 
icSf  edited  and  published  by  Joel  E.  Hendricks,  a  self-taught  mathema- 
tician. Mr.  Hendricks  did  the  printing  of  the  journal  himself.  It  con- 
tinued until  November,  1883.  No  previous  journal  of  mathematics  in 
this  country  had  been  published  regularly  for  so  long  a  time  as  this. 
Its  long  life  and  beneficial  influence  are  due  to  a  very  great  extent  to 

*  Analyst,  Vol.  II,  No.  5,  p.  131,  Dea  Moines,  Iowa. 


280  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

the  untiring  energy  and  self-sacrificing  interest  of  its  editor.  Its  dis- 
continuance, after  nine  years,  was  not  due  to  want  of  support,  but  to 
tlie  failing  health  and  strength  of  Mr.  Hendricks.  At  first  it  appeared 
monthly,  afterward  bi-monthly. 

The  list  of  contributors  included  the  most  prominent  teachers  of 
mathematics  in  this  country.  The  names  were  no  longer  those  found  in 
the  Mathematical  Miscellany  or  Cambridge  Miscellany.  A  new  gener- 
ation of  workers  had  come. 

As  in  previous  periodicals,  so  in  this,  a  great  part  of  each  number 
was  devoted  to  problems.  Though  the  solution  of  problems  is  the  low- 
est form  of  mathematical  research,  it  is,  nevertheless,  important,  not  for 
its  scientific,  but  for  its  educational  value.  It  induced  teachers  to  look 
beyond  the  text-book  and  to  attempt  work  of  their  own.  The  Analyst 
bears  evidence,  moreover,  of  an  approaching  departure  from  antiquated 
views  and  methods,  of  a  tendency  among  teachers  to  look  into  the  history 
and  philosophy  of  mathematics  and  to  familiarize  themselves  with  the 
researches  of  foreign  investigators  of  this  century.  Thus,  discussions 
regarding  the  fundamental  principles  of  the  differential  calculus  were 
carried  on.  Levi  W.  Meech  gave  an  "  Educational  Testimony  Concern- 
ing the  Calculus ; "  W.  D.  Wilson,  of  Cornell,  gave  "A  E"ew  Method  of 
Finding  Differentials ; "  Joseph  Ficklin,  of  Missouri,  showed  how  one 
might  "  find  the  differential  of  a  variable  quantity  without  the  use  of  in- 
finitesimals or  limits  5 "  C.  H.  Judson,  of  South  Carolina,  gave  a  valu- 
able "investigation  of  the  mathematical  relations  of  zero  and  infinity," 
which  displayed  the  wholesome  effects  of  the  study  of  such  authors  as 
De  Morgan.  Judson  dealt  powerful  blows  against  the  reckless  reason- 
ing that  had  been  in  vogue  so  long,  but,  during  an  occasional  unguarded 
moment,  he  was  hit  by  his  opponents  in  return.  De  Volson  Wood,  of 
the  Stevens  Institute,  and  Simon  ^"ewcomb,  of  Washington,  discussed 
the  doctrine  of  limits. 

Another  subject  considered  in  the  Analyst  was  the  possibility  of  an 
algebraic  solution  of  equations  of  the  fifth  degree.  A.  B.  Nelsou,  pro- 
fessor of  mathematics  in  Centre  College,  Danville,  Ky.,  translated  from 
the  German  an  article  written  in  18G1  by  Adolph  Von  Der  Schulen- 
burg,  entitled,  "Solution  of  the  General  Equation  of  the  Fifth  De- 
gree." The  translation  and  publication  of  it  seem  to  have  been  called 
forth  by  a  statement  of  W.  D.  Henkle  in  the  Educational  Notes  aud 
Queries,  to  the  effect  that  proofs  of  the  impossibility  of  such  a  solution 
had  been  given  by  Abel  and  Wantzel.  Kelson's  paper  was  followed  by 
a  translation  from  Serret's  Gours  WAlgebre  Super ieure,  by  Alexander 
Evans,  of  Elkton,  Md.,  of  Wantzel's  "  Demonstration  of  the  Impossi- 
bility of  Eesolving  Algebraically  General  Equations  of  a  Degree  Higher 
than  the  Fourth."  Evans  also  contributed  the  (non-algebraic)  "'Solu- 
tion of  the  Equation  of  the  Fifth  Degree,"  translated  from  the  Theory 
of  Elliptic  Functions  of  Briot  and  Bouquet.  W.  E.  Heal,  of  Wheeling, 
Ind.,  followed  with  an  article  pointing  out  the  error  in  Schulenburg's 


MATHEMATICAL  JOURNALS.  281 

solution.  One  miglit  have  supposed  that  this  question  had  now  come  to 
a  rest,  but  not  so.  About  two  years  later  T.  S.  B.  Dixon,  of  Chicago, 
thought  he  had  found  a  solution,  and  he  i)ublished  it  in  the  Analyst, 
but,  in  the  next  number,  he  stated  that  he  had  discovered  "  the  weak 
link  in  the  chain"  of  its  logic. 

Of  the  articles  on  modern  higher  mathematics,  we  mention  the  "Brief 
Account  of  the  Essential  Features  of  Grassmann's  Extensive  Algebra," 
by  W.  W.  Beman,  of  Ann  Arbor;  "Symmetrical  Functions,  etc.,"  and 
"  Eecent  Eesults  in  the  Study  of  Linkages,"  by  W.  W.  Johnson,  and 
an  article  on  determinants  by  0.  A.  Yan  Yelaer,  of  the  University  of 
Wisconsin. 

Among  the  historical  papers  is  the  very  complete  and  interesting 
"  Historical  Sketch  of  American  Mathematical  Periodicals,"  by  David 
S.  Hartj  of  Stonington,  Conn. ;  a  "  History  of  the  Method  of  Least 
Squares,"  by  M.  Merriman.  Merriman  also  published  Bobert  Adrain's 
second  proof  of  the  principle. 

Among  other  articles  of  interest  are  "Multisection  of  Angles,"  and 
"A  Singular  Value  of  ;r,"  by  J.  W.  Nicholson,  of  The  Louisiana  State 
University,  at  Baton  Eouge.  The  latter  article  was  commented  upon 
by  W.  W.  Johnson,  then  professor  of  mathematics  in  St.  John's  College, 
Annapolis,  Md.,  who  was  a  frequent  and  most  gifted  contributor  to  the 
Analyst.  Asaph  Hall  wrote  on  comets  and  meteors,  George  E.  Perkins 
on  indeterminate  problems,  E.  B.  Seitz  on  probability.  Other  impor- 
tant contributors  were  Daniel  Kirkwood,  David  Trowbridge,  Artemas 
Martin,  and  G.  W.  Hill. 

Well  known  among  the  mathematical  public  of  America  is  Artemas 
Martin.  .  Before  speaking  of  his  two  periodicals  we  shall  briefly  sketch 
his  life.  This  gives  us  at  the  same  time  an  opportunity  of  mentioniug 
many  publications  which,  though  not  purely  mathematical,  contained  a 
mathematical  department.  We  can  think  of  few  American  periodicals 
of  the  last  thirty  years,  paying  any  considerable  attention  to  elementary 
mathematics,  for  which  Dr.  Martin  has  not  been  a  contributor.  Dr. 
Martin  was  born  in  1835.  In  1869  he  moved  to  Erie  County,  Pa.,  where 
he  lived  on  a  farm  for  fifteen  years,  engaged  as  a  market- gardener.  He 
is  almost  wholly  self-taught.  His  leisure  moments  were  devoted  to  the 
study  of  the  "bewitching  science."  Through  the  influence  of  the  Hon. 
W.  L.  Scott,  Member  of  Congress  from  Erie,  Martin  was  appointed,  in 
]885,  librarian  in  the  office  of  the  U.  S.  Coast  and  Geodetic  Survey. 
He  has  a  large  private  library  containing  a  very  fine  collection  of 
American  books  on  mathematics.  When  the  writer  was  in  Washington 
he  enjoyed  the  great  privilege  of  examining  this  collection  and  of  seeing 
many  a  quaint  and  curious  volume  of  great  rarity. 

Martin  began  his  mathematical  career  when  ia  his  eighteenth  year,  by 
contributing  solutions  to  the  Pittsburg  Almanac  and  soon  afterward 
contributed  problems  to  the  "  Eiddler  Column "  of  the  Philadelphia 
Saturday  Evening  Post,  and  was  one  of  the  leading  contributors  for 


282  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

twenty  years.  In  1864  lie  began  contributing  problems  and  solutions 
to  ClarWs  Scliool  Visitor,  afterward  the  School-day  Magazine,  published 
in  Philadelphia.  In  June,  1870,  he  took  charge  of  the  "  Stairway  De- 
partment "  as  editor,  the  mathematical  part  of  which  he  had  conducted 
for  some  years  before.  In  1875  he  was  chosen  editor  of  the  department 
of  higher  mathematics  in  the  Normal  Monthly ,  published  at  Millersville, 
Pa.,  by  Edward  Brooks.  The  Monthly  was  discontinued  in  1876.  In 
this  journal  he  published  a  series  of  sixteen  articles  on  Diophantine 
analysis.  He  contributed  to  the  mathematical  department  of  the  Illi- 
nois Teacher  (1865-67) ;  the  Iowa  Instructor,  1865 ;  the  National  Edu- 
cator, Kutztown,  Pa.;  the  Yates  County  Chronicle,  a  weekly  paper  of 
New  York,  the  mathematical  department  of  which  was  edited  by  Samuel 
H.  Wright;  Barneses  Educational  Monthly  ;  the  Maine  Farmers''  Almanac; 
Educational  Notes  and  Queries,  edited  and  published  by  W.  D.  Heukle, 
of  Ohio.  Dr.  Martin  published  thirteen  articles  on  "average"  in 
Wittenherger,  from  1876  to  1880  inclusive.  The  mathematical  depart- 
ment of  this  was  edited  by  William  Hoover,  afterward  professor  of 
mathematics  in  the  Ohio  University  at  Athens.  Martin's  name  is 
familiar  also  to  the  readers  of  the  School  Visitor,  a  journal  started  in 
1880,  and  edited  and  published  monthly  by  John  S.  Eoyer  in  Gettys- 
burg, Darke  County,  Ohio;  of  the  Davenport  Monthly,  Davenport, 
Iowa ;  and  of  The  Bizarre,  conducted  by  S.  0.  and  L.  M.  Gould,  in  Man- 
chester, N.  H.  All  these  journals  devoted  a  portion  of  their  space  to 
mathematics,  and  to  all  these  Dr.  Martin  contributed.  The  mathe- 
matics they  contained  were  of  course  of  an  elementary  kind.  He  con- 
tributed also  to  English  journals  on  elementary  mathematics.  Besides 
the  above  periodicals  we  mention  the  Railroad  Gazette  (New  York  and 
Chicago),  which  contained  problems  in  applied  mathematics;  the 
Mathematician,  edited  by  Koyal  Cooper,  1877,  and  utterly  worthless ; 
and  the  Wheel,  New  York,  1868,  of  which  only  one  number  ever  ap- 
peared, in  which  the  question  was  discussed  how  many  revolutions 
upon  its  own  axis  a  wheel  will  make  in  rolling  once  around  a  fixed 
wheel  of  the  same  size.* 

In  the  spring  of  1877  Artemas  Martin  issued  the  first  number  of  his 
Mathematical  Visitor,  which  he  still  continues  to  publish  annually. 
"Although  he  has  never  served  an  hour  as  apprentice  in  a  printing  office 
to  learn  the  art  preservative,  he  has  done  all  the  type-setting  for  his 
publications,  except  that  for  the  first  three  numbers  of  the  Visitor,  and 
has  printed  all  the  numbers  of  the  Visitor  except  the  first  five  on  a  self- 
inking  lever  press  only  6J  x  10  inches  inside  of  chase.  The  numbers  of 
the  Visitor  he  has  printed  himself  have  been  pronounced  by  competent 
judges  to  be  as  fine  specimens  of  mathematical  printing  as  have  ever 
been  executed.    The  Magazine  he  puts  in  type  and  gets  the  presswork 

*  For  a  more  complete  list  of  journals  containing  mathematical  departments,  see 
The  Bizarre,  Notes  and  Queries,  Volume  V,  No.  12,  December,  1888. 


MATHEMATICAL  JOURNALS.  283 

done  at  a  printing  office,  as  his  press  is  too  small  to  safely  print  it, 
although  he  printed  one  number  on  it."* 

Of  the  Visitor  generally  six  hundred  copies  have  been  printed.  The 
list  of  contributors  exceeds  one  hundred.  In  the  introduction  Dr.  Mar- 
tin says:  *'It  was  stated  nearly  three-quarters  of  a  century  ago  that 
the  learned  Dr.  Hutton  declared  that  the  Ladies'  Diary  had  produced 
more  mathematicians  in  England  than  all  the  mathematical  authors  in 
that  kingdom."  The  aim  of  the  Visitor  is,  if  possible,  to  reach  similar 
results  in  this  country.  It  is  devoted  to  the  solution  of  problems. 
They  deal  more  particularly  in  Diophantine  analysis,  average,  and  prob- 
ability. 

In  January,  1883,  Dr.  Martin  issued  the  first  number  of  the  Mathe- 
matical  Magazine^  which  is  published  quarterly.    It  was  intentionally 
made  more  elementary  than  the  Analyst  of  Mr.  Hendricks  or  the  An- 
nals of  Mathematics.    It  was  devoted  mainly  to  arithmetic,  algebra, 
geometry,  and  trigonometry..    One  of  the  features  is  the  solution  and 
discussion  "of  such  of  the  problems  found  in  the  various  text-books  in 
use  as  are  of  special  interest,  or  present  some  difficulty."    Many  of  the 
articles  found  in  the  Magazine  and  Visitor  came  from  the  pen  of  the 
editor  himself.    liTumerous  different  proofs  of  the  Pythagorean  propo- 
sition were  given  in  the  former,  of  which  we  may  mention  one  by  James 
A.  Garfield.    It  was  taken  from  a  magazine  of  1876  or  1877,  and  was 
found  pasted  on  a  fly-leaf  of  an  old  geometry.    It  resembles  somewhat 
the  old  Hindoo  proof.    Dr.  G.  B.  Halsted  contributed  several  articles  on 
the  prismoidal  formula.    J.  W.  Nicholson  gave  a  "  universal  demonstra- 
tion" of  the  binomial  theorem,  without,  however,  giving  a  moment's 
thought  to  the  question  of  convergency,  whenever  the  series  is  infinite. 
William  Hoover  gave  an  interesting  little  article  on  the  history  of  the 
algebraic  notation.    David  S.  Hart  wrote  on  the  history  of  the  theory 
of  numbers,  including  the  indeterminate  and  Diophantine  analysis.    He 
also  contributed  several  articles  on  the  subject  last  mentioned.    A 
somewhat  lengthy  discussion  was  carried  on,  on  the  usefulness  of  log- 
arithms, by  P.   H.  Philbrick,  professor  of  engineering  at   the  State 
University  of  Iowa,  and  H.  A.  Howe,  professor  of  mathematics  at  the 
University  of  Denver.    The  former  attempted  to  show  that  the  use  of 
logarithms  greatly  augmented  the  labor  of  "  numerical  computation  " 
and  led  to  very  erroneous  results.    Some  of  the  calculations  in  the  mag- 
azine in  which  numerical  answers  are  carried  to  twenty  or  more  decimal 
places  have  no  value,  either  educational  or  scientific.    The  names  of  the 
contributors  for  the  magazine  were  about  the  same  as  for  the  Visitor. 

To  show  the  good  that  elementary  journals  like  this  may  do,  we 
give,  as  an  example,  the  career  of  E.  B.  Seitz.  He  passed  his  boyhood 
on  a  farm,  and  afterward  pursued  a  mathematical  course  of  two  years 
at  the  Ohio  Wesley  an  University.  In  1872  ho  began  contributing 
problems  proposed  in  the  "  Stairway "  department  of  the  School-day 

*The  Buffalo  Express,  August  29, 1886. 


284  TEACHING   AND   HISTORY   OP  MATHEMATICS. 

Magazine  conducted  by  Dr.  Martin.  His  energies  were  stimulated,  and 
lie  became  a  leading  contributor  to  our  periodicals.  He  astonished  liis 
friends  by  his  skill  in  solving  difficult  problems,  and  their  admiration 
for  his  talents  became  almost  unbounded.  His  mathematical  mind  had 
received  the  first  stimulus  from  our  elementary  periodicals.  Had  he 
not  died  in  the  prime  of  life,  he  might  have  done  good  original  work, 
provided  he  had  begun  to  look  higher  than  merely  to  the  solution  of 
difficult  problems  in  our  elementary  journals.  The  solving  of  problems 
is  very  beneficial  at  first,  but  it  becomes  a  waste  of  time  if  one  confines 
himself  to  that  sort  of  work.  The  solution  of  problems  is  not  a  high 
form  of  mathematical  research,  and  should  serve  merely  as  a  ladder  to 
more  ambitious  efforts. 

Another  journal  devoted  mainly  to  the  solution  of  problems  is  the 
Scliool  Messenger,  now  called  the  Mathematical  Messenger,  edited  and 
published  bi-monthly  by  G.  H.  Harvill,  at  Ada,  La.  One  of  the  ablest 
contributors  to  it  is  J.  W.  iSTicholson,  professor  of  mathematics  at  the 
Louisiana  State  University.  The  Messenger  commenced  February, 
1884. 

The  Annals  of  Mathematics  is  the  continuation,  under  a  new  name  and 
different  form,  of  the  Analyst.  It  is  edited  and  published  at  the  Uni- 
versity of  Virginia  by  Prof.  Ormond  Stone  and  Prof.  William  M.  Thorn- 
ton. It  is  of  somewhat  higher  grade  than  the  Analyst,  though  more 
elementary  than  the  American  Journal  of  Mathematics.  It  contains 
more  articles  on  mathematical  astronomy  and  other  subjects  of  applied 
mathematics  than  the  American  Journal.  Our  distinguished  math- 
ematical astronomer,  G.  W.  Hill,  contributes  several  articles  in  his 
specialty.  Profs.  Asaph  Hall,  E.  S.  "Woodward,  H.  A.  Howe,  and  Wil- 
liam M.  Thornton  contribute  various  articles  on  applied  mathematics. 
Professor  Oliver,  of  Cornell,  has  several  papers,  one  on  "A  Projective 
Eelation  among  Infinitesimal  Elements,"  and  another  on  the  "  General 
Linear  Differential  Equation."  Prof.  W.  W.  Johnson  writes  on 
"  Glaisher's  Factor  Tables,"  the  "  Distribution  of  Primes,"  and  other  sub- 
jects. Professor  Halsted  gives  his  demonstration  of  Descarte's  theorem 
and  Euler's  theorems.  The  name  of  Bohannan,  now  professor  at  Ohio 
University,  appears  often.  Prof.  O.  H.  Mitchell,  of  Marietta  College, 
discusses  the  equation  of  the  second  degree  in  two  variables.  Prof.  E. 
H.  Graves  has  geometrical  articles ;  William  E.  Heal  writes  on  repetends; 
S.  T.  Moreland,  on  the  momental  ellipsoid;  J.  F.  McCullogh,  on  Eolle's 
theorem.  A  little  space  in  each  number  is  devoted  to  the  proposing 
and  solving  of  problems.  The  list  of  contributors  is  too  large  to  be 
given  here  in  full. 

When  Professor  Sylvester  became  actively  connected  with  the  Johns 
Hopkins  University,  in  1877,  the  university  established  the  American 
Journal  of  Mathematics,  for  the  publication  of  original  research  in  pure 
and  applied  mathematics.  It  was  the  design  that  this  should  not  be  a 
journal  devoted  to  the  publication  of  solutions  to  problemSj  but  that  it 


MATHEMATICAL   JOURNALS.  285 

should  be  of  so  high  a  grade  as  to  command  a  place  by  the  side  of  the 
best  European  journals  of  mathematics.  It  is  a  source  of  pride  to  us 
that  this  great  aim  has  been  reached.  The  American  Journal  of  Mathe- 
matics is  to-day  as  well  known  and  as  highly  respected  in  Europe  as  ia 
America.  Among  its  contributors  are  found  not  only  the  leading  scien- 
tists of  America,  but  also  such  foreign  investigators  as  Cayiey,  Clifford, 
Crofton,  Faa  de  Bruno,  Frankland,  De  Gasparis,  Glashan,  Hammond, 
Hermite,  Kempe,  Lipschitz,  Loudon,  Lucas,  MacMahon,  Muir,  Petersen, 
Poincar^,  Roberts,  Weichold,  and  G.  P.  Young. 

The  subject  which  has  received  most  attention  in  the  American  Jour- 
nal of  Mathematics  has  been  Modern  Higher  Algebra.  The  contribu- 
tions of  Sylvester  on  this  subject  loom  large.  In  Volume  I  is  found 
"a  somewhat  speculative  paper"  entitled,  "An  Application  of  the 
New  Atomic  Theory  to  the  Graphical  Eepresentation  of  the  Invariants 
and  Covariants  of  Binary  Quantics,"  followed  by  appendices  and  notes 
relating  to  various  special  points  of  the  theory.*  Sylvester  contributed 
various  memoirs  on  binary  and  ternary  quantics,  including  papers  by 
himself,  with  the  aid  of  Dr.  Franklin,  containing  tables  of  the  numer- 
ical generating  functions  for  binary  quantics  of  the  first  ten  orders,  and 
for  simultaneous  binary  quantics  of  the  first  four  orders,  etc.  The  list 
of  his  articles  is  too  extensive  to  be  mentioned  here.  Since  his  return 
to  England  he  has  contributed  to  the  Journal  a  series  of  "  Lectures 
on  the  Theory  of  Eeciprocants,"  reported  by  J.  Hammond. 

The  larger  number  of  American  contributions  are  from  persons  who 
were,  or  still  are,  connected  with  the  Johns  Hopkins  University,  either 
as  teachers  or  students.  Dr.  W.  E.  Story,  of  the  Johns  Hopkins  Uni- 
versity, has  written  on  "  llTon-Buclidean  Trigonometry,"  "Absolute 
Classification  of  Quadratic  Loci,  etc.,"  and  other^  chiefly  geometrical, 
subjects.  Dr.  T.  Craig  has  contributed  numerous  papers,  mainly  on 
the  theory  of  functions  and  differential  equations.  Dr.  F.  Franklin  has 
aided  Professor  Sylvester  in  the  preparation  of  papers,  and  has  also 
made  various  independent  contributions.  After  the  return  to  England 
of  Professor  Sylvester,  Professor  N"ewcomb  became  editor-in-chief.  His 
valuable  articles  have  been  noticed  elsewhere.  Among  the  contribu- 
tors who  were  once  students  at  the  Johns  Hopkins  University,  but 
are  now  not  connected  with  it,  are  B.  W.  Davis,  W.  P.  Durfee,  G.  S. 
Ely,  G.  B.  Halsted,  A.  S.  Hathaway,  O.  H.  Mitchell,  W.  I.  Stringham, 
C.  A.  Yan  Yelzer,  A.  L.  Daniels,  C.  Yeneziani,  D.  Barcroft,  and  J.  0. 
Fields.  The  Journal  has  two  lady  contributors,  Mrs.  C.  Ladd  Frank- 
lin, of  Baltimore,  and  Miss  0,  A.  Scott,  of  Bryn  Mawr  College.  The 
great  memoir  on  "Linear  Associativ^e  Algebra,"  by  Benjamin  Peirce, 
was  published  in  the  American  Journal  of  Mathematics  5  also  articles 
by  his  son,  C.  S.  Peirce,  on  the  "Algebra  of  Logic  "  and  the  "  Ghosts  in 
Diffraction  Spectra."  Papers  on  applied  mathematics  have  been  written 
by  Professor  Eowland,  of  the  Johns  Hopkins  University,  and  George 

"  rWe  Professor  Cayiey 's  article  oa  Professor  Sylyesterin  Nature,  Januarys,  1889. 


286  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

William  Hill,  of  the  Kaatical  Almanac  Office.  Mr.  Hill  lias  done  ad- 
mirable work  in  mathematical  astronomy.  For  his  researches  on  the 
lunar  theory,  published  in  the  American  Journal,  and  for  other  astro- 
nomical papers,  published  elsewhere,  he  was  awarded  the  gold  medal  of 
the  Royal  Astronomical  Society,  in  1887.*  Among  the  writers  for  the 
American  Journal  is  Prof.  W.  W.  Johnson,  of  the  U.  S.  Naval  Acad- 
emy at  Annapolis.  He  is  also  a  frequent  contributor  to  leading  Euro- 
pean journals,  and  commands  a  place  among  the  very  foremost  of 
American  mathematicians.  In  the  list  of  American  writers  to  the 
Journal  are  H.  T.  Eddy,  J.  W.  Gibbs,  E.  McGlintock,  A.  W.  Phillips, 
J.  Hagen,  B.  W.  Hyde,  H.  B.  Fine,  and  others  of  no  less  power  and 
originality. 

The  U.  S.  Coast  and  Geodetic  Survey. 

In  giving  the  origin  of  the  U.  S.  Coast  Survey  it  is  desirable  to  begin 
with  a  sketch  of  the  preliminary  training  of  its  first  superintendent, 
Ferdinand  B>.  Hassler.  He  was  born  in  the  town  of  Aarau,  Switzer- 
land, in  1770.  He  was  sent  to  the  University  of  Bern  to  study  law,  but 
he  soon  drifted  into  mathematics  and  became  a  favorite  pupil  of  Prof. 
John  G.  Tralles.t  Tralles  and  Hassler  undertook  the  topographical 
survey  of  the  Canton  of  Bern.  In  1791  a  base-line  was  measured,  and 
a  net  of  triangles  established.  The  instruments  on  hand  being  found 
insufficient  for  long  distances  new  ones  were  ordered  from  Eamsden, 
in  London.  On  the  receipt  of  these,  in  1797,  the  survey  was  resumed, 
but  soon  discontinued.  The  conquering  armies  of  the  French  came 
marching  into  Switzerland.  The  feeble  republic  was  forced  to  submit 
to  the  dictatorial  orders  of  the  war  minister  of  France,  which  required, 
among  other  things,  that  the  places  then  occupied  by  the  Swiss  engi- 
neers should  be  vacated  and  filled  by  French.  A  swarm  of  sixty 
French  engineers  appeared,  but  soon  disappeared  without  accomplish- 
ing anything.  The  revolutionary  tendencies  of  the  times  and  the  un- 
settled state  of  the  <50untry  induced  Hassler  to  quit  Switzerland.  His 
fatherland  seemed  to  bear  no  roses  for  him.  Having  landed  in  Phila- 
delphia, in  October,  1805,  he  formed  the  acquaintance  of  Prof.  Eobert 
Patterson  and  Mr.  Garnet,  of  New  Brunswick,  to  whom  he  showed  his 
mathematical  books  and  instruments. 

About  this  time  Congress  was  considering  the  feasibility  of  a  survey 
of  the  coasts  and  harbors.  Professor  Patterson  sent  to  President  Jef- 
ferson a  sketch  of  Hassler's  scientific  career  in  Switzerland,  and  Mr. 
Clay,  the  Eepresentative  from  Philadelphia,  in  1806,  asked  Hassler 
whether  he  would  be  willing  to  undertake  the  survey,  in  case  that  the 

*  Vide  Monthly  Notices  on  the  Eoyal  Astronomical  Society,  Vol.  XL VII,  No.  4, 
February,  1887. 

t  Translation  from  the  German  of  Memoirs  of  Ferdinand  Rudolph  Hassler,  by  Emil 
Zschokke,  published  in  Aarau,  Switzerland,  1877,  with  Supplementary  Documents, 
published  in  Nice,  1883. 


THE  U.  S.  COAST  AND  GEODETIC  SURVEY.       287 

Government  should  decide  upon  one.  Mr.  Hassler  was,  of  courae,  willing. 
The  law  authorizing  the  survey  was  passed  in  February,  1807.  Hass- 
ler received  one  of  the  twelve  circulars  which  were  sent  to  scientific 
men  for  plans  of  the  contemplated  survey.  By  the  direction  of  Presi- 
dent Jefferson,  a  commission  (formed,  it  appears,  of  the  very  gentlemen 
who  had  proposed  plans,  excepting  Mr.  Hassler)  examined  the  various 
plans  at  Professor  Patterson's,  in  Philadelphia.  They  rejected  their 
own  projects  and  recommended  to  the  President  the  one  suggested  by 
Mr.  Hassler.  The  survey  proposed  by  him  was  of  a  kind  that  had  never 
been  previously  attempted  in  this  country;  it  was  to  be  a  triangulation, 
and  the  sides  of  the  triangles  were  to  be  from  ten  to  sixty  miles  in 
length,  such  as  had,  at  that  time,  just  been  executed  in  France  and  was 
in  progress  in  England.  The  project  was  quite  in  advance  of  the  sci- 
ence of  our  country.  It  was  fortunate  for  us  that  a  man  of  Mr.  Hass- 
ler's  learning,  ability,  and  mechanical  ingenuity  was  available  to  the 
Government.  He  had,  meanwhile,  been  appointed  by  Jefferson  pro- 
fessor at  West  Point.  This  position  he  resigned  after  three  years,  and 
accepted  the  professorship  of  mathematics  at  Union  College,  Schenect- 
ady, N.  Y.  Politics  delayed  the  work  of  the  survey.  The  first  thing 
to  be  done  was  to  procure  the  necessary  instruments.  In  1811  Hassler 
was  sent  to  England  by  our  Government  to  direct  the  manufacture  of 
euitable  instruments.  Shortly  after  his  arrival  in  Great  Britain  the  War 
of  1812  broke  out,  and  he  was  four  years  in  London,  in  the  disagreeable 
position  of  an  alien  enemy,  and  half  the  time  left  by  our  Government 
without  compensation.  He  returned  to  this  country  in  1814,  with  a 
splendid  collection  of  instruments,  which  had  cost  nearly  forty  thousand 
dollars. 

In  August,  1816,  a  formal  agreement  between  the  Government  and 
Mr.  Hassler  was  reached,  to  the  effect  that  he  should  undertake  the 
execution  of  the  survey.  He  immediately  entered  upon  the  preliminary 
steps  of  reconnoitering  and  the  numerous  collateral  experiments  neces- 
sary for  such  a  survey.  Two  preliminary  base-lines  were  measured : 
One  in  Kew  Jersey,  in  the  rear  of  the  Highlands,  on  liforth  Eiver,  and 
nearly  six  miles  in  length;  the  other  on  Long  Island,  and  of  about  five 
miles.  Down  to  the  year  1818  eleven  stations  were  occupied,  forming 
the  elements  of  124  triangles. 

To  a  scientific  man,  familiar  with  the  many  preliminary  details  which 
are  indispensable  to  accurate  scientific  work,  but  which  do  not  always 
appear  in  the  ultimate  results,  the  progress  which  Hassler  was  makiug 
would  have  seemed  highly  satisfactory.  Congress,  however,  was  dis- 
pleased. In  April,  1818,  Mr.  Hassler  received  official  notice  that  he  was 
suspended,  accompanied  with  the  remark  that  the  little  progress  hitherto 
made  in  the  work  had  caused  general  dissatisfaction  in  Congress.  Pos- 
sibly the  feeling  on  the  part  of  American  engineers  against  this  for- 
eigner because  he  had  been  preferred  to  one  of  them  had  something  to 
do  with  this  suspension.    To  Hassler  this  was  a  ver j  severe  blow ;  his 


288  TEACHING    AND    HISTORY    OF   MATHEMATICS. 

brightest  hopes  seemed  dashed  into  fragments.  A  year  or  two  later  he 
prepared  a  defense  of  himself.  He  wrote  an  account  of  his  plans  and 
methods  and  published  it  in  the  Philosophical  Transactions  of  Phila- 
delphia (Vol.  Ill,  New  Series,  1825).  By  this  article  he  hoped  to  vindi- 
cate his  schemes.  It  attracted  the  attention  of  scientific  men  every- 
where. It  "was  reviewed  by  leading  astronomers  in  Europe — Bessel, 
Struve,  Schumacher,  F^russac,  Francceur,  Krusenstern,  and  others — all 
agreeing  that  Mr.  Hassler's  plans  were  good,  and  testifying  to  his  in- 
ventive genius  for  solving  the  difficulties  of  the  Coast  Survey,  as  well 
as  to  the  certainty  that  his  plans,  if  carried  out,  would  lead  to  success. 
Bessel  was  certainly  a  competent  judge,  for,  in  addition  to  his  theo- 
retical knowledge,  he  had  had  experience  in  geodetic  work  in  Germany. 
He  had  words  of  only  the  highest  praise  for  Hassler's  scheme.* 

After  his  suspension  from  the  survey,  Hassler  engaged  in  various 
occupations.  For  a  while  he  was  a  farmer  in  northern  K'ew  York.  He 
afterward  went  to  Jamaica,  Long  Island,  and  then  to  Eichmond,  Va., 
giving  lessons  in  mathematics  to  sons  of  prominent  men.  While  in 
Eichmond  he  published  his  Elements  of  the  Geometry  of  Planes  and 
Solids,  1828.  His  Elements  of  Analytic  Trigonometry  appeared  in  1826. 
Subsequently  he  published  an  Arithmetic,  Astronomy,  and  Logarithms 
and  Trigonometric  Tables,  with  introductions  in  five  languages. 

After  twelve  years  in  rural  retreat,  Hassler  was  recalled  to  official 
activity.  He  became  United  States  gauger,  and  then  was  intrusted, 
from  1830  to  1832,  with  the  difficult  mission  of  regulating  the  standards 
of  weights  and  measures  throughout  the  United  States,  which  at  that 
time  were  very  various.    • 

In  1828  the  question  of  the  Coast  Survey  was  again  agitated.  The 
Secretary  of  the  Navy  reported  to  Congress  favorably  on  Hassler's 
work,  which  had  been  suspended  so  suddenly  ten  years  previously. 
The  Secretary  said  that  "  he  [Hassler]  had  accomplished  all  that  was 
possible  in  so  short  a  time."  In  1832  Mr.  Hassler  was  reinstated,  with 
the  title  of  "  Superintendent  of  the  United  States  Coast  Survey." 

In  the  interval  from  1818  to  1832  nothing  of  permanent  value  had 
been  accomplished.  Attempts  had  been  made  to  survey  portions  of  the 
coast,  under  the  direction  of  the  Navy  Department,  but  there  had  been 
no  general  or  connected  survey.  The  charts  prepared  had  been  expen- 
sive and  unsafe,  and  not  very  creditable  to  the  country. 

In  1832  began  the  most  successful  and  most  famous  period  in  Mr. 
Hassler's  life.  Though  sixty-two  years  ol  d,  there  still  glowed  in  him  the 
fire  of  youth.  The  survey  was  begun  with  vigor.  He  had  a  traveling 
carriage  prepared  for  him,  which  conveyed  him  rapidly  to  all  parts  of 
the  survey.  In  this  carriage  he  could  seat  himself  at  a  writing  table 
or  dispose  himself  for  sleep.  The  work  was  prosecuted  accordingto  the 
plans  first  laid  out  by  him.  He  labored  under  the  great  disadvantage 
of  having  no  skilled  assistants.    His  corps  of  workmen  had  all  to  be 

*  Fide  Silliman's  Journal,  Vol.  IX,  p.  225. 


THE  U-  S.  COAST  AND  GEODETIC  SURVEY.       289 

trained  and  educated  to  the  refined  methods  which  he  was  introducing. 
The  work  of  the  survey  had  to  be  systematized.  It  continued  under  his 
direction  until  the  time  of  his  death,  in  1843.  He  left  the  work  well 
advanced  between  Narragansett  Bay  and  Cape  Henlopen,  and  the  sur- 
vey sufficiently  organized  in  all  its  varied  details.  His  course  was,  how- 
ever, frequently  criticised  in  Congress,  and  it  was  not  always  easy  to 
get  the  necessary  appropriations. 

Mr.  Hassler  was  very  self-confident  and  independent.  This  was  one 
cause  of  the  occasional  opposition  to  him.  Though  not  conceited,  he 
was  conscious  of  his  superiority  over  the  great  mass  of  men  with  whom 
he  came  in  contact  in  Washington.  The  following  anecdote  is  charac- 
teristic of  him :  At  one  time  the  cry  of  "retrenchment  and  reform" 
had  become  popular,  and  a  newly  appointed  Secretary  of  the  Treasury 
thought  he  could  not  signalize  his  administration  more  aptly  than  by 
reducing  the  large  salary  of  the  superintendent.  He  sent  for  Mr.  Hass- 
ler and  said,  "My  dear  sir,  your  salary  is  enormous;  you  receive  $6,000 
per  annum — an  income,  do  you  know,  quite  as  large  as  that  of  the  Sec- 
retary of  State."  "  True,"  replied  Hassler,  "  precisely  as  much  as  the 
Secretary  of  State  and  quite  as  much  as  the  Chief  of  the  Treasury ;  but 
do  you  know,  Mr.  Secretary,  that  the  President  can  make  a  minister  of 
State  out  of  anybody;  he  can  make  one  even  out  of  you,  sir;  but  if  he 
can  make  a  Hassler,  I  will  resign  my  place." 

Hassler's  successor  was  Alexander  Dallas  Bache,  a  great-grandson  of 
Franklin  and  a  graduate  of  West  Point.  He  exercised  a  very  marked 
influence  over  the  progress  of  science  among  us.  He  graduated  at  the 
head  of  his  class,  and  the  great  expectations  that  were  then  entertained 
of  him  have  been  fully  realized.  For  eight  years  he  devoted  himself  to 
physical  science,  while  professor  at  the  University  of  Pennsylvania, 
and  gained  a  wide  reputation.  The  Coast  Survey  made  rapid  progress 
under  his  management.  Congress  began  to  show  better  appreciation  of 
this  sort  of  work,  and  made  more  liberal  appropriations.  This  enabled 
him  to  adopt  a  more  comprehensive  scheme.  Instead  of  working  only 
at  one  locality,  as  had  been  done  previously,  he  was  able  to  begin 
independent  surveys  at  several  places  at  once,  each  section  employing 
its  own  base.  He  proposed  eight  sections,  which  number  was  increased 
on  the  annexation  of  Texas,  and  again  on  the  annexation  of  California. 

Two  of  the  most  important  improvements  of  modern  geodesy  were 
perfected  and  brought  into  use  at  the  beginning  of  Bache's  superin- 
tendency,  namely,  Mr.  Talbott's  method  of  determining  latitudes  and 
the  telegraphic  method  of  determining  longitude.  Various  other  re- 
finements in  every  branch  of  work  were  introduced.  Systematic  obser- 
vations of  the  tides,  a  magnetic  survey  of  the  coast,  and  the  extension 
of  the  hydrographic  explorations  into  the  Gulf  Stream  were  also  insti- 
tuted by  Bache. 

Having  extended  the  scope  of  the  Survey,  Bache  needed  a  greater 
number  of  assistants,  but  the  supply  was  not  wanting.  Says  Prof.  T. 
881— Ko.  3 19 


290  TEACHING    AND   HISTORY   OF   MATHEMATICS. 

H.  Safford,*  "  he  found  available  for  its  higher  geodetic  works  a  number 
of  West  Point  officers,  of  whom  T.  J.  Lee  was  one,  and  Humphreys,  af- 
terward chief  engineer  of  the  Army,  another.  One  of  the  leaders  in 
practical  astronomy  of  the  topographical  engineers  was  J.  D.  Graham ; 
and  the  work  which  had  been  done  by  that  corps  upon  the  natTonal  and 
State  boundaries  had  familiarized  a  good  many  Army  officers  with  field 
astronomy  and  geodesy. 

"Bache,  who  had  been  out  of  the  Army  nearly  twenty  years  employed 
his  great  organizing  and  scientific  capacity  in  training  the  Coast  Survey 
corps  (including  detailed  Army  officers)  into  practical  methods  for  its 
various  problems ;  and  the  connection  between  the  West  Point  officers 
and  the  able  young  civilians,  who  are  now  the  veterans  of  the  Survey, 
was  extremely  wholesome. 

"  Lee  prepared  a  work  (Tables  and  Formulae)  which  has  served  an 
excellent  purpose  in  bridging  the  gap  between  theory  and  practice ; 
especially  for  the  last  generation  of  West  Point  officers. " 

Graduates  of  West  Point  are  now  more  closely  employed  in  military 
and  other  public  duty,  and  are  no  longer  employed  in  the  Coast  Survey. 

The  work  of  the  Survey  was  interrupted  by  the  Civil  War.  Soon 
after  its  close  Bache  died  (1867).  Benjamin  Peirce,  his  successor  in  the 
superintendency,  said  of  him  :  "  What  the  Coast  Survey  now  is,  he 
made  it.  It  is  his  true  and  lasting  monument.  It  will  never  cease  to 
be  the  admiration  of  the  scientific  world.  *  *  *  It  is  only  necessary 
conscientiously  and  faithfully  to  follow  in  his  foot-steps,  imitate  his  ex- 
ample, and  develop  his  plans  in  the  administration  of  the  Survey." 

Under  Peirce,  the  survey  of  the  coasts  was  pushed  with  vigor,  and 
it  rapidly  approached  completion.  He  proposed  the  plan  of  connecting 
the  survey  on  the  Atlantic  Coast  with  that  on  the  Pacific  by  two  chains 
of  triangles,  a  northern  and  a  southern  one.  This  project  received  the 
sanction  of  Congress,  and  thus  the  plan  of  a  general  geodetic  survey 
of  the  whole  country  was  happily  inaugurated. 

Benjamin  Peirce's  successor  on  the  Coast  Survey  was  Carlile  Pollock 
Patterson.  He  was  a  graduate  of  Georgetown  CoUege,  Kentucky,  and 
had  for  many  years  previous  to  this  appointment,  in  1874,  been  connected 
with  the  Survey  as  hydrographic  inspector.  Under  him  the  extension 
of  the  Survey  into  the  interior  of  our  country,  as  inaugurated  by  Peirce, 
was  continued.  By  the  completion  of  this  work  this  country  will  con- 
tribute its  fair  share  to  the  knowledge  of  the  figure  of  the  earth,  which 
has  hitherto  been  derived  entirely  from  the  researches  of  other  nations. 
On  account  of  this  extension,  the  name,  "  U.  S.  Coast  Survey,  "  was 
changed,  in  1879,  to  "  U.  S.  Coast  and  Geodetic  Survey. " 

Patterson  died  in  1881,  and  Julius  Erasmus  Hilgard  became  his  suc- 
cessor. Hilgard  was  born  in  Zweibriicken,  Bavaria,  came  to  this  coun- 
try at  the  age  of  ten,  and  at  the  age  of  twenty  was  invited  by  Bache  to 
become  one  of  his  assistants  on  the  Survey.    Hilgard  soon  came  to  be 

^Mathematical  Teachings,  p.  6. 


THE  U.  S.  COAST  AND  GEODETIC  SURVEY.        291 

recognized  for  great  ability  and  skill,  and  rose  to  the  position  of  assist. 
ant  in  charge  of  the  Office  in  Washington.  He  held  the  superintend- 
ency  from  1881  to  1885,  when  he  resigned.  His  work  consisted  chiefly 
of  researches  and  discussions  of  results  in  geodesy  and  terrestrial 
physics,  and  in  the  perfecting  of  the  methods  and  instruments  em- 
ployed. The  superintendency  was  next  intrusted  to  Frank  M.  Thorn, 
who  was  succeeded  in  July,  1889,  by  T.  C.  Meudenhall,  who  now  fills 
t!je  office. 

The  work  of  the  U.  S.  Coast  Survey  has  been  carried  on  with  great 
efficiency  from  its  very  beginning,  and  reflects  great  credit  upon  Amer- 
ica. In  making  the  computations  for  the  Survey,  the  method  of  least 
squares  for  the  adjustment  of  observations  has  found  extended  appli- 
cation. Valuable  papers  on  this  subject  by  Bache  and  Schott  have 
been  printed  in  the  reports  of  the  XJ.  S.  Coast  Survey.*  Charles  A. 
Schott  graduated  at  the  Polytechnic  School  in  Carlsruhe,  came  to  this 
country  in  1848,  and  has  since  that  time  been  an  efficient  worker  on  the 
U.  S.  Coast  Survey.    He  is  now  chief  of  the  computing  division. 

It  will  be  remembered  that  interesting  researches  on  least  squares 
had  been  made  quite  early  in  this  country  by  Eobert  Adrain.  Benjamin 
Peirce  invented  a  criterion  for  the  rejection  of  doubtful  observations.t 
It  proposes  a  method  for  determining,  by  successive  approximations, 
whether  or  not  a  suspected  observation  may  be  rejected.  Tables  are 
needed  for  its  application.  Objections  have  been  made  to  its  use,  be- 
cause it  "involves  a  contradiction  of  reasoning."!  The  criterion  is 
given  by  Ohauvenet  in  his  Method  of  Least  Squares  (1864),  and  has 
been  used  to  some  extent  on  the  U.  S.  Coast  Survey,  but  it  has  found 
no  acceptance  in  Europe.  Chauvenet  gives  an  approximate  criterion 
of  his  own  for  the  rejection  of  one  doubtful  observation,  which  is  de- 
rived, he  says,  "  directly  from  the  fundamental  formula  upon  which  the 
whole  theory  of  the  method  of  least  squares  is  based."  But  this  cri- 
terion, too,  has  been  criticised  as  being  "  troublesome  to  use,  and  as 
based  on  an  erroneous  principle."  Stone,  in  England,  offered  still  an- 
other criterion  for  the  rejection  of  discordant  observations,  but  Glaisher 
pronounces  it  untrustworthy  and  wrong.  No  criterion  has  yet  been 
given  which  has  met  with  general  acceptance.  Indeed,  Professor  New- 
comb  considers  it  impossible  that  such  a  one  should  ever  be  invented. 
Says  he  (in  his  second  paper  mentioned  below) :  "  We  here  meet  the 
difficulty  that  no  positive  criterion  for  determining  whether  an  observa- 
tion should  or  should  not  be  treated  as  abnormal  is  possible.  Several 
attempts  have  indeed  been  made  to  formulate  such  a  criterion,  the  best 
known  of  which  is  that  of  Peirce." 

*  See  reports  for  the  years  1850,  '55,  '56,  '58,  '61,  '64,  '66,  '67,  '75. 

t  Gould's  Astronomical  Journal,  Vol.  II,  pp.  161-3. 

t  See  Prof.  Mansfield  Merriman'a  article  in  the  Transactions  of  the  Conneotiont 
Academy,  containing  a  list  of  writings  relating  to  the  method  of  least  squares  and 
the  theory  of  the  accidental  errors  of  observation^  which  comprises  408  titles  by  193 
authors. 


292  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

Valuable  papers  on  least  squares  have  been  contributed  in  this  coun- 
try by  G.  P.  Bond,*  of  Harvard ;  Simon  Newcomb,  t  C.  S.  Pierce,  |  and 
Truman  H.  Safford.§  The  text-books  on  this  subject  generally  used  in 
our  schools  are  those  of  Ohauvenet,  Merriman,  and  T.  W.  Wright. 

*  "On  the  use  of  E quivalent  Factors  in  the  Method  of  Least  Squares,"  Memoirs  Ameri- 
can Academy,  Vol.  VI,  pp.  179-212. 

t "  A  Mechanical  Representation  of  a  Familiar  Problem,"  Monthly  Notices  of  the 
Astronomical  Society,  London,  Vol.  XXXIII,  pp.  573-'4;  "A  Generalized  Theory  of 
the  Combination  of  Observations  so  as  to  Obtain  the  Best  Results,"  American  Jour- 
nal of  Mathematics,  Vol.  VIII. 

X  "On  the  Theory  of  Errors  of  Observations,"  Report  U.  S.  Coast  Survey,  1870,  pp. 
200-224. 

J  "  On  the  Method  of  Least  Squares/'  Proceedings  American  Academy,  Vol.  XI. 


IV. 

THE  MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME. 

The  mathematical  teaching  of  the  last  ten  years  indicates  a  "rup- 
ture" with  antiquated  traditional  methods,  and  an  "  alignment  with  the 
march  of  modern  thought."  As  yet  the  alignment  is  by  no  means  recti- 
fied. Indeed  it  has  but  barely  begun.  The  "rupture"  is  evident  from 
the  publication  of  such  works  as  Newcomb's  series  of  mathematical  text- 
books, recent  publications  on  the  calculus,  the  appearance  of  such  alge- 
bras as  those  of  Oliver,  Wait,  and  Jones,  Phillips  and  Beebe,  and  Yan 
Yelzer  and  Slichter ;  of  such  geometries  as  Halsted's  "  Elements  "  and 
"Mensuration;"  of  such  trigonometries  as  Oliver,  Wait,  and  Jones's; 
of  CarlPs  Calculus  of  Variations ;  Hardy's  Quaternions ;  Peck's  and 
Hanus's  Determinants ;  W.  B.  Smith's  Co-ordinate  Geometry  (employ- 
ing determinants) ;  Craig's  Linear  Differential  Equations. 

Determinants  and  quaternions  have  thus  far  generally  been  offered 
as  elective  studies,  and  have  formed  a  crowning  pinnacle  of  the  mathe- 
matical courses  in  colleges.  It  is  certainly  very  doubtful  whether  this 
is  their  proper  place  in  the  course.  It  seems  quite  plain  that  the  ele- 
ments of  determinants  should  form  a  part  of  algebra,  and  should  be 
taught  early  in  the  course,  in  order  that  they  may  be  used  in  the  study  of 
co-ordinate  geometry.  What  place  should  be  assigned  to  qua,ternions  is 
not  quite  so  plain.  Prof.  De  Volson  Wood  introduces  their  elements  in 
his  work  on  co-ordinate  geometry.  The  professors  of  Cornell  have  not 
taught  quaternions  directly  for  some  years,  but  are  convinced  that  most 
students  derive  more  benefit  by  a  mixed  course  in  matrices,  vector  ad- 
dition and  subtraction,  imaginaries,  and  theory  of  functions.  The  early 
introduction  of  determinants  seems  more  urgent  than  that  of  quaterni- 
ons. We  think,  however,  that  great  caution  should  be  exercised  in  in- 
corporating either  of  these  subjects  in  the  early  part  of  mathematical 
courses.  Those  universities  and  colleges  which  are,  as  yet,  not  strong 
enough  to  maintain  a  high  and  rigid  standard  of  admission,  and  whose 
students  enter  the  Freshman  class  with  only  a  very  meagre  and  super- 
ficial knowledge  of  the  elements  of  ordinary  algebra,  would  find  the  in- 
troduction of  determinants  and  imaginaries  as  Freshman  studies  a 
hazardous  innovation.  One  of  the  very  first  considerations  in  mathe- 
matical teaching  is  thoroughness.  In  the  past  the  lack  of  thoroughness 
has  poisoned  the  minds  of  the  American  youth  with  an  utter  dislike  and 
bitter  hatred  of  mathematics.  Whenever  a  subject  is  not  well  under- 
stood, it  is  not  liked;  whenever  it  is  well  understood,  it  is  generally 

liked., 

293 


294  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

There  is  almost  always  some  one  author  whose  text-books  reach 
very  extended  i)opularity  among  the  great  mass  of  schools.  Such  au- 
thors were  Webber,  Day,  Davies,  and  Loomls.  If  we  were  called  upon 
to  name  the  writer  whose  books  have  met  with  more  wide-spread  circu- 
Jation  during  the  last  decennium  than  those  of  any  other  author  we 
should  answer,  Wentworth.  Mr.  Wentworth  was  born  in  Wakefield, 
N.  H.,  fitted  for  college  at  Phillips  Exeter  Academy,  graduated  at  Har- 
vard College  in  1858,  and  then  returned  to  Phillips  Exeter  Academy, 
where  he  has  been  ever  since.  He  had  for  instructors  in  mathematics, 
at  the  academy,  Prof.  Joseph  G.  Hoyt,  afterward  chancellor  of  the 
Washington  University  in  St.  Louis  ;  and,  in  college,  Prof.  James  Mills 
Peirce.  "  The  characteristics  of  my  books,"  says  Mr.  Wentworth, "  are 
due  to  what  I  have  found  from  a  long  experience  is  absolutely  necessary 
in  order  that  a  pupil  of  ordinary  ability  might  master  the  subject 
of  his  reading.  To  learn  by  doing,  and  to  learn  one  step  thoroughly 
before  the  next  is  attempted,  constitute  pretty  much  the  whole  story." 
In  point  of  scientific  rigor  Wentworth's  books  are  superior  to  the  popu- 
lar works  of  preceding  decades.  It  seems  to  us  that  the  book  most 
liable  to  criticism  is  his  Elementary  Geometry  (old  edition).  He  has 
been  greatly  assisted  in  the  writing  of  his  books  by  leading  teachers 
from  different  parts  of  our  country.  Some  of  the  books  bearing  his 
name  are  almost  entirely  the  work  of  other  men. 

It  is  to  be  hoped  that  the  near  future  will  bring  reforms  in  the  mathe- 
matical teaching  in  this  country.  We  are  in  sad  need  of  them.  From 
nearly  all  our  colleges  and  universities  comes  the  loud  complaint  of  in- 
efficient preparation  on  the  part  of  students  applying  for  admission  j 
from  the  high  schools  comes  the  same  doleful  cry.  Errors  in  mathe- 
matical instruction  are  committed  at  the  very  beginning,  in  the  study 
of  arithmetic.  Educators  who  have  studied  the  work  of  Prussian 
schools  declare  that  our  results  in  elementary  instruction  are  far  infe- 
rior. Says  President  C.  K.  Adams,  of  Cornell  University:  *  "  In  the 
lowest  grades  of  schools  our  inferiority  seems  to  me  to  be  very  marked. 
The  results  of  the  earliest  years  of  the  European  course,  I  mean  those 
devoted  to  teaching  the  boy,  say  from  the  time  he  is  nine  years  of  age 
until  he  is  fourteen,  when  compared  with  the  fruits  of  the  courses  pur- 
sued during  the  corresponding  years  in  the  average  American  school, 
are  immeasurably  superior."  President  Adams  institutes  a  comparison 
between  Brooklyn  and  Berlin  schools.  Speaking  of  a  Brooklyn  boy  of 
fifteen,  he  remarks :  "  In  the  first  place  it  must  be  said  that  he  has  had 
forced  upon  him  six  hours  a  week  in  arithmetic,  during  the  whole  of 
the  seven  primary  grades.  Then  on  emerging  from  the  primary  school, 
and  coming  into  the  grammar  school,  he  is  required  to  take  an  average 
of  four  hours  a  week  in  the  same  study,  during  all  the  eight  grades. 
That  is  to  say,  during  the  whole  of  the  boy's  career  in  school,  from  the 

*New  England  Association  of  Colleges  and  Preparatory  Schools;  Addresses  and 
proceedings  at  the  Annual  Meeting,  1888,  p.  24. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.    295 

time  lie  is  seven  until  lie  is  fifteen,  he  has  devoted  no  less  than  five  hours 
a  week  of  recitations  to  the  study  of  arithmetic  alone.  If  we  deduct 
the  hours  devoted  to  reading,  penmanship,  and  music,  we  find  that  five- 
elevenths  of  what  remains  is  devoted  to  arithmetic.  Making  no  de- 
ductions, and  including  the  hours  devoted  to  the  elementary  work  re- 
quiring no  preparation  whatever,  we  find  that  arithmetic  occupies  in 
the  class-room  considerably  more  than  one-fourth  of  all  the  student's 
time,  during  the  whole  of  seven  or  eight  years." 

This  statement  is  applicable  with  equal  force  to  probably  all  our 
schools.  The  fact  is  that  the  study  of  arithmetic  has  been,  in  one 
sense,  greatly  overdone  in  this  country.  The  most  melancholy  thought 
in  this  connection  is  that,  after  all,  our  boys  and  girls  acquire  only  a 
deficient  knowledge  of  this  subject.  Persons  who  had  opportunity  for 
comparison  assure  us  that  the  American  boy  does  not  "figure"  as 
rapidly  and  accurately  as  the  German  boy. 

If  the  above  assertions  be  true,  then  it  behooves  the  American 
teacher  to  inquire  wherein  the  foreign  methods  of  teaching  excel  his 
own.  In  some  circles  the  study  of  pedagogy  has  not  been  popular. 
This  apathy  is,  we  think,  partly  due  to  the  influence  of  some  of  our  nor- 
mal schools.  Many  of  our  normal  schools  have  been  conducted  very 
efficiently,  but  others  have  had  teachers  in  their  faculties  who  lacked 
breadth  and  depth  of  scholarship,  and  who  brought  the  study  of  peda- 
gogy into  disrepute  by  their  narrowness  and  their  lack  of  elasticity  in 
the  application  of  methods.  This  aversion  to  the  study  of  theories  of 
teaching  is  now  happily  disappearing.  Our  universities  and  colleges 
are  beginning  to  establish  chairs  of  pedagogy. 

Improvements  in  the  teaching  of  arithmetic  might  probably  be 
effected  by  the  general  introduction  of  some  such  method  as  that  of 
Grube.  The  first  complete  exposition  of  this  method  was,  we  believe, 
published  in  this  country  by  F.  L.  Soldan,  formerly  principal  of  the  St. 
Louis  Normal  School.    It  seems  to  gain  ground  here  every  year. 

A  most  valuable  and  suggestive  monograph  on  mathematical  teach- 
ing has  been  written  by  Prof.  T.  H.  Safford,  of  Williams  College. 
Professor  Safford  is  an  advocate  of  the  heuristiG  method  of  teaching. 
Grube  is  the  representative  of  this  in  arithmetic.  The  method  employed 
by  Spencer  in  his  little  book  on  Inventional  Geometry  is  similar  to  the 
heuristic,  if  not  identical  with  it.  The  heuristic  is,  in  general,  the 
method  in  which  the  pupil's  mind  does  the  work.  It  is  a  slow  method. 
Thus,  Grube  considers  the  numbers  from  1  to  10  sufficient  to  engage 
the  attention  of  a  child  (of  six  or  seven  years)  during  the  first  year  of 
school.  "In regard  to  extent,  the  scholar  has  not,  apparently,  gained 
very  much — he  knows  only  the  numbers  from  1  to  10.  But  he  knows 
them."*  The  Germans  "  make  haste  slowly,"  but  in  elementary  educa- 
tion they  beat  us  in  the  race.  Geometry,  like  arithmetic,  should  be 
taught  sparingly  at  a  time,  but  for  many  years  in  succession.    Profcs- 

*  Grube's  method,  by  F.  Louis  Soldan,  p.  21. 


296 


TEACHING  AND   HISTORY  OP  MATHEMATICS. 


sor  Safiford  strongly  recommends  the  parallelism  of  the  two  main 
mathematical  subjects — arithmetic  including  algebra,  and  geometry 
including  trigonometry  and  conic  sections.  Thereby  the  study  of 
algebra  and  geometry  can  be  extended  over  a  longer  period  of  time.  Ac- 
cording to  his  ideal  programme  of  study,  primary  arithmetic  is  accom- 
panied by  notions  of  form  and  drawing ;  arithmetic  through  rule  of 
three,  by  rudiments  of  geometry ;  universal  arithmetic  and  simple  equa- 
tions, by  one  or  two  books  in  plane  geometry ;  algebra  through  quad- 
ratics, by  plane  geometry ;  advanced  algebra,  by  solid  geometry,  conic 
sections,  plane  trigonometry,  etc. 

"  Of  course  this  programme  is  somewhat  variable,  but  the  main  prin- 
ciple, that  a  course  of  arithmetic  must  run  parallel  with  one  of  geometry 
from  the  beginning  of  a  school  course  to  the  end,  is  one  which  is  laid 
down  by  the  best  educators  since  Pestalozzi's  time."* 

In  order  to  enable  the  writer  to  give  a  view  of  the  present  condi- 
tion of  mathematical  teaching,  the  Bureau  of  Education  sent  to  various 
universities,  colleges,  normal  schools,  academies,  institutes,  and  high 
schools,  a  printed  letter  with  a  series  of  questions  to  be  answered.  We 
give  a  list  of  the  institutions  which  sent  in  replies,  and  state  the  re- 
sults as  fully  as  our  space  will  permit.! 


STATISTICS  ILLUSTRATING  THE  PRESENT  CONDITION  OF  MATHEMAT- 
ICAL INSTRUCTION  IN  THE   UNITED  STATES. 

(a)  Universities  and  Colleges. 


Name  of  instittition. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

1 

■University  of  Alabama . . . 

Tuscaloosa,  Ala 

T.W.  Palmer 

Professor  of  mathematics. 

f. 

Spring  Hill  CoUege 

Central  Female  College. . . 

Mobile,  Ala 

A.  S.  Wagner 

S.B.  Foster...... 

Do. 

3 

Tuscaloosa,  Ala 

President. 

4 

Alabama  Polytechnic  In- 
stitute, Agricultural  and 
Mechanical  College. 

Auburn,  Ala 

0.  D.Smith 

Professor  of  mathematics. 

5 

HuntsviUe  Female  College 

Huntsville,  Ala 

A.  B. Jones 

President. 

6 

Talladega  College 

Talladega,  Ala 

Jesse  Bailey 

Principal. 

7 

Alabama  Conference  Fe- 
male College. 

Feskegie,  Ala 

John  Massey 

President. 

8 

Pierce  Christian  College.. 

College  City,  Cal... 

D.  E.  Hughes 

Professor  of  mathematics, 
astronomy,  and  civil  en- 
gineering. 

9 

St.  Ignatius  College 

San  Francisco,  Cal. . 

T.C.Leonard 

Teacher  of  higher  mathe- 
matics. 

10 

State  Agricultural  College 

Fort  Collins,  Colo... 

V.E.Stolbraiid... 

Professor  of  mathematics 
and  m  asier  of  literary  sci- 
ence. 

n 

University  of  Denver 

Denver,  Colo 

H.A.Howe 

Professor  of  mathematics 
and  astronomy. 

*  Monograph  on  Mathematical  Teaching  by  T.  H.  Saflbrd,  p.  44. 

t  From  this  list  are  omitted  some  few  reports  which  were  sent  iu  too  late  for  insertion,  or  which  did 
not  give  the  name  of  the  institution  or  the  person,  reporting,  or  -which  were  illegible. 


MATHEMATICAL   TEACHING  AT   THE   PRESENT  TIME.         297 
(a)  Universities  and  Colleges— Continued. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

12 

ITniversity  of  Colorado .. . 

Boulder,  Colo 

LM.DeLong 

Professor  of  mathematics. 

13 

State  School  of  Minea 

Golden,  Colo 

Paul  Meyer 

Do. 

14 

Storra  Agricultural  School 

Storrs,  Conn 

W.  P.  Washburn.. 

Professor  of  chemistry  and 
mathematics. 

in 

Trinity  College  ...... 

Hartford,  Conn 

F  S  T  ntViPT 

Professor  of  matliematics 
and  astronomy. 

16 

University  of  Dakota 

Vermillion,  Dak 

L.S.Hulburt 

Professor  of  mathematics. 

17 

Dakota  School  of  Mines  . . 

Rapid  City,  Dak.... 

L.  L.  Conant 

Do. 

18 

Georgetown  College 

"Washington,  D.  C .  -  - 

J.F.Dawson 

Professor  of  physics  and 
mechanics. 

19 

Kational  Deaf-Mute  Col- 
lege. 

....do 

f  Joseph  C.  Gordon 
<  A.  G.  Draper 

Professor  of  mathematics. 

Assistant   professor   of 

mathematics. 

20 

Howard  University 

..-.do 

G.W.Cook 

Tutor  in  mathematics. 

21 

De  Land  University 

DeLandjFla 

R.Gentry 

22 

Seminary  West  of  the  Su- 
wannee Eiver. 

Tallahassee,  Fla  — 

G.M.Edgar 

President  and  professor  of 
mathematics  and  natural 
science. 

23 

Florida  State  Agricultural 
College. 

Lake  City,  Fla 

L.  H.  Orleman 

Professor  of  mathematics. 

24 

Bowdon  Colleo'e  .......... 

C.  0  Stubbs 

Do 

25 

North  Georgia  Agricult- 
ural College. 

Dahlonega,  Ga 

W.S.Wilson 

Do. 

?fi 

Cave  Spring,  Ga 

E.T.Whatley  .... 
W.  Rutherford 

?7 

University  of  Georgia 

?8 

Eureka,  111 

G.A.Miller 

Do. 

29 

Illinois  State  Normal  Uni- 
versity. 

Normal,  111 

J.W.Cook 

Instructor  in  mathematics. 

30 

Lombard  University 

Galesburg,  HI 

J.V.N.Standish.. 

Professor  of  mathematics 
and  astronomy. 

31 

M'Kendree  College 

Lebanon,  El 

A.G.  Jepson 

Professor  of  mathematics. 

V 

German- English  College.. 
Lincoln  University 

Fr.  Schaub 

33 

Lincoln,  HI 

J.  W.  P.  Buchanan 

Professor  of  mathematics. 

34 

Lake  Forest  University  . . 

Lake  Forest,  HI 

M.  McNeill 

Professor  of  mathematics 
and  astronomy. 

^'y 

University  of  Illinois 

f  S.  W.  Shattuck  . . 
(S.H.  Peabody 

Professor  of  mathematics. 

Regent  (president). 

?6 

Jacksonville,  111 

Napersville,  HI 

J.H.Pratt 

Ph.  D. 

37 

North-Western  College . . . 

H.  F.Kletzing.... 

Professor  of  mathematics. 

38 

Indiana  University 

Wabash  College ...... 

Bloomington,  Ind  . . . 
Crawfordsville,  Ind . 

Do. 

39 

J.  Norris 

Do. 

40 

Earlham  College 

Richmond,  Ind 

W.B.Morgan 

Do. 

41 

Eose  Polytechnic   Insti- 
tute. 

De  Pau w  University 

Franklin  College 

Terre  Haute,  Ind  . . . 

C.  A.  Waldo 

Do. 

4' 

43 

Franklin,  Ind 

Covington,  Ind 

E.  J.Thompson... 
O.E.  Coffin 

Professor  of  mathematics. 

44 

Indiana  Normal  College  . . 

4R 

Hanover  College 

Hanover,  Ind 

Iowa  City,  Iowa 

P.L.Morse 

L.G.Weld 

Professor  of  mathematics. 

46 

State  University  of  Iowa. 

Acting  professor  of  math- 

ematics. 

47 

University  of  Des Moines. 

Des  Moines,  Iowa... 

T.M.Blakslee.... 

Ph.  D.,  Tale,  1880. 

iS 

Oskaloosa  College 

Oskaloosa,  Iowa 

J.  A.  Beattie 

President. 

£9 

Upper  Iowa  University  .. 

Fayette,  Iowa. ...... 

J.W.Bresell 

Do. 

298  TEACHING  AND   HISTOEY   OF   MATHEMATICS, 

(a)  Universities  and  Colleges— Continued. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

eo 

Oawego  College  for  Young 
Ladies. 

Oswego,  Kan 

S.H.Johnson 

Principal. 

•ii 

University  of  Kansas 

Ottawa  University 

E.Miller 

Professor  of  mathematics. 

52 

Ottawa,  Kan 

M.L."Ward 

Professor  of  mathematics 

and  political  science. 

53 

Bethany  College  and  Nor- 
mal Institute. 

Lindsborg,  Kan 

"W.A.Granville  .. 

Professor  of  mathematics. 

54 

Washburn  College 

Topeka,  Kan 

P.Mc  Vicar 

President. 

65 

Kansas  State  Agricultural 
College. 

Manhattan,  Kan 

D.E.Lantz 

Professor  of  mathematica. 

56 

"West  Kentucky  Classical 
and  Normal  College. 

SouthCarrollton,Ky 

"W.C.Gaynor 

President. 

57 

Millersburg  Pemale  Col- 
lege. 

Millersburg,  Ky 

C.Pope.. 

Do. 

5fi 

Berea,  Ky 

P.D.Dodge 

Acting  professor  of  math- 

ematics. 

59 

Eminence,  Ky 

H.  Boring 

Teacher   of  mathematics, 

Normal  School. 

Latin,  and  Greek. 

f>n 

Bowling  Green,  Ky . 

"W.  A.  Ohmcliain.. 

President. 

61 

Kentucky  Classical   and 
Business  College. 

North  Middletown, 

S.  "W.Pearcy 

Do. 

62 

Hamilton  Female  College 

Lexington,  Ky 

J.  "W.Porter 

Professor  of  mathematics 
and  Latin. 

63 

G."W.Thigpen.... 

Professoi?  of  mathematics. 

College. 

64 

Mount  St.  Mary's  College. 

Emmitsburg,  Md  . . . 

J.A.Mitchell 

Professor. 

65 

"Western  Maryland  College 

"Westminster,  Md. . . 

"W.  R.  McDaniel  .. 

Professor  of  mathematics. 

66 

Baltimore  City  College  — 

Baltimore,  Md 

"W.Elliott 

Principal. 

67 

Johns  Hopkins  University 

....do  

S.  Newcomb 

Professor  of  mathematics 

and  astronomy. 

68 

U.  S.  Naval  Academy 

Annapolis,  Md 

"W.  W.  Hendriok- 
son,  J.  M.  nice. 

Professors  of  mathematics. 

69 

Maryland  Agricultural 
College. 

Agricultural  College 
P.  0.,  Md. 

H.E.AIvord 

President. 

70 

St.  John's  College 

Annapolis,  Md 

J.W.Cain 

J.  N.  Hart 

Professor  of  mathematics. 

71 

Maine  State  College  of  Ag- 
riculture and  Mechanic 

Instructor  in  mathematics. 

Arts. 

79 

Colby  University 

"Waterville,  Me 

L.  E.  "Warren 

Professor  of  mathematics. 

73 

East    Maine  Conference 
Seminary. 

Bucksport,  Me 

A.  F.Chase 

Principal. 

74 

Brunswick,  Me 

Lewistoii,  Me 

W.A.Moody 

J.M.Kand 

Professor  of  mathematics. 

75 

Bates  College 

Do. 

76 

Agricultural  College. 1 

Amherst,  Mass 

C.  D.  "Warner 

Professor  of  mathematics 
and  physics. 

77 

"Wesleyan  Academy 

"Wilbraham,  Mass... 

B.  S.  Annis 

Instructor  in  mathematicB. 

78 

The  Society  for  the  Colle- 
giate   Instruction     of 
"Women. 

Cambridge,  Mass  -  - . 

A.  Gilman 

Secretary. 

7t 

Northampton,  Mass. 
Aubumdale,  Mass .  - 

E.P.  Cushing 

L.M.Packard 

Teacher  of  mathematios. 

80 

Instructor  in  mathematicu. 

81 

Swain  Free  School 

New  Bedford,  Mass . 

A.  In  graham 

Master. 

82 

Thayer  AcadMny 

Braintiee,  Mass 

C.A.Pitkin, 

Master  of  mathematics  and 

physics. 

MATHEMATICAL  TEACHmCJ  AT  THE  PRESENT  TIME.        299 
(a)  Universities  and  Colleges — Continued. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

HI 

Amherst,  Mass  ..... 
Boston,  Mass 

"W.C.Esty 

J.D.Runkle 

84 

Massacliusetts   Institute 

Professor  of  mathematics. 

of  TecliHology. 

85 

■Williaton  Seminary 

Eastharapton,  Mass. 

"W.  C.Boyden 

Instructor  in  mathematics. 

86 

"WiUiamstown,  Mass 
Worcester,  Mass 

T.  H.  SafFord 

Professor  of  astronomy. 
Professor  of  higher  mathe- 

87 

Polytechnic  Institute 

J.E.Sinclair 

matics. 

88 

Mount  Holyoke    Female 
Seminary. 

South  Hadley,  Mass . 

E."W.  Bard  well... 

Director  of  observatory. 

89 

Michigan  Mining  School. . 

Houghton,  Mich 

R.M.  Edwards  ... 

Professor  of  mining    and 
engineering. 

00 

Battle  Creek  College 

Battle  Creek,  Mich.. 

J.  H.  Haughey 

Mathematical  department 

ni 

Adrian,  Mich 

Hillsdale,  Mich 

Minneapolis,  Minn . . 

G.B.McElroy  .... 

A.  E.  Haynes 

J.F.Downey 

Chairman  of  the  faculty. 

q") 

HUlsdale  College 

93 

Minnesota  State  Univer- 

Professor of  mathematics 

sity. 

and  astronomy. 

04 

Hamlin  e    University    of 
Minnesota. 

Hamline,  Minn 

E.F.Mearkle 

Professor  of  mathematics. 

95 

"Washington  University  . . 

St.  Louis,  Mo 

CM.  "Woodward.. 

Do. 

96 

Kansas  City  Ladies'  Col- 
lege. 

Independence,  Mo . . 

J.  M.  Chaney  ..... 

President. 

97 

Missouri  State  University. 

Columbia,  Mo 

"W.B.Smith 

Professor  of  mathematics 
and  astronomy. 

98 

School  of  Mines,  Univer- 
sity of  Missouri. 

Eolla,Mo 

TV.  H.  Echols 

Professor  of  applied  math- 

ematics. 

99 

College  of  the  Christian 
Brothers. 

St.  Louis,  Mo 

Brother  Paulian . . 

President. 

100 

"William  Jewell  College  . . . 

Liberty,  Mo 

J.E.Clark 

A.  F.  Amadon 

Professor  of  mathematics. 

101 

Springfield,  Mo 

Professor  of  mathematics 

and  physics. 

102 

Cooper  Normal  College  . . . 

Daleville,  Misa 

T.F.McBeath.... 

President. 

103 

Agricultural  and  Mechan- 

Stark ville.  Miss 

H.C.Davis 

Acting  professor  of  mathe- 

ical College. 

matics. 

104 

University  of  Mississippi. 

University  P.    0., 
Miss. 

CM.  Sears 

Professor  of  mathematics. 

105 

A.  B.  Show 

Librarian. 

106 

University  of  Nehraska. . . 

Lincoln,  Nebr 

H.  E.  Hitchcock  . . 

Professor  of  mathematics. 

107 

New  Hampshire  College 

Hanover,  N.  H 

C.H.Pettee 

Do. 

of  Agriculture  and  Me- 

chanic Arts. 

108 

"Wake  Forest  College 

Wake  Forest,  N.C.. 

L.R.  Mills 

Professor  of  pore  mathe- 
matics. 

109 

University  of  North  Caro- 
lina. 

Chapel  Hill,  N.  C  .. 

J.  L.  Love. ........ 

Associate  professor  of 

mathematics. 

110 

Randolph    County, 

N.C. 
Burke  County,  N.C. 

J.  M.  Bandy 

Professor  of  mathematics. 

111 

Rutherford  College 

R.  L.  Abernethy  . . 

President. 

112 

College    of    the    Sacred 

Vineland.N.J 

P.  O'Connor 

Professor  of  mathematics. 

Heart. 

113 

Niagara  University 

Niagara,  N.Y 

E.  A.Antill 

Do. 

114 

Schenectady,  N.  T  .. 
New  York,  N.  Y  . . . . 

B.  H.  Ripton 

Do. 

U5 

Columbia  College 

r  W.  G.  Peck 

iT.S.Fiske 

Do. 
Tutor  in  mathematics. 

300  TEACHING  AND   HISTORY   OF   MATHEMATICS, 

(a)  Universities  and  CoLLKGES—Continued. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

116 

University  of  Eochester  . . 

Eochester,  N.  T 

G.D.Olds 

Professor  of  mathematics. 

in 

St.  Lawrence  University.. 

Canton.N.Y 

H.  Priest 

Professor  of  mathematics 

and  physics. 

118 

The  College  of  tlie  City  of 

New  York,  N.Y.... 

A.S.Webb 

President. 

New  York. 

119 

Syracuse  University 

Syracuse,  N.Y 

J.  E.  French 

Professor  of  mathematics. 

120 

U.  S.  Military  Academy . . . 

"West  Point,  N.  T  . . . 

E.W.Bass 

Do. 

121 

Packer   Collegiate  Insti- 

Brooklyn, N.Y 

W.L.  C.Stevens.. 

Professor  of  mathematics 

tute. 

and  physics. 

199 

Brooklyn   Collegiate  and 
Polytechnic  Institute. 

....do 

E.  Sheldon 

Professor  of  pure  mathe* 

matics. 

1?3 

Ohio  University 

Athens,  Ohio 

WiEiam  Hoover  . . 

Professor  of  mathematics. 

124 

Ohio  State  University 

Columbus,  Ohio 

E.  D.Bohannan... 

Professor  of  mathematics 
and  astronomy. 

125 

Miami  University 

Oxford,  Ohio 

J.V.Collins 

Do. 

126 

Case   School  of  Applied 
Science. 

Cleveland,  Ohio 

C.Staley 

President. 

197 

Hiram,  Ohio 

C.Bancroft 

Professor  of  mathematics 

and  astronomy. 

T'S 

OberUn,  Ohio 

H.  C.  King 

Associate     professor     of 

mathematics. 

129 

Deniaon  University 

Granville,  Ohio 

J.  L.  Gilpatrick  . . . 

Professor  of  mathematics. 

ISO 

Marietta,  Ohio 

O.H.MitcheU 

Professor  of  mathematics 

and  astronomy. 

ni 

Akron,  Ohio 

C.  S.  Howe 

Do. 

132 

University  of  Cincinnati.. 

Cincinnati,  Ohio 

H.  T.Eddy 

Professor  of  mathematics, 
civil  engineering,  and  as- 
tronomy. 

133 

Pacific  University 

Forest  Grove,  Oro- 

"W.N.Ferrill 

Professor  of  mathematics. 

134 

The   State    Agricultural 
College  of  Oregon. 

Corvallis,  Oregon  . . . 

J.D.Letcher 

Professor  of  mathematics 
and  engineering. 

ISt 

Dickinson  College 

Bryn  Mawr  College 

Carlisle,  Pa 

F.Durell 

Professor  of  mathematics. 

136 

Bryn  Mawr,  Pa 

Charlotte  A.  Scott. 

Associate     professor      of 

mathematics. 

137 

Pardee  Scientific  Depart- 
ment of  Lafayette  Col- 

J.G.Fox  

graphical  engineering. 

lege. 

138 

Lehigh  University 

South  Bethlehem,  Pa. 

C.  L.  Doolittle 

Professor  of  mathematics. 

139 

Swarthmore  College 

Swarthmore,  Pa 

S.  J.  Cunningham  . 
f  Isaac  Sharpless. . 
(  Frank  Morley  - . . 

Do. 

Professor  of  mathematics. 

140 

Haverford  College 

Haverford,  Pa 

Instructor  in  mathematics. 

141 

Muhlenberg  College 

Allentown,  Pa 

D.  Garber 

Professor  of  astronomy. 

142 

Central  Pennsylvania  Col- 

New Berlin,  Pa 

H.E.  Kelly 

Professor  of  mathematics. 

lege. 

143 

Brown  University 

Providence,  R.  I 

N.F.Davis 

Assistant    professsor    of 
mathematics. 

144 

Furman  University 

Greenville,  S.  C 

C.  H.  Judson 

Professor  of  mathematics 
and   mechanical   philos- 
ophy. 

145 

University  of  South  Caro- 

Columbia, S.C 

E.'W.  Davis 

Professor  of  mathematics 

lina. 

and  astronomy. 

148 

Columbia  Female  College. 

....do 

J.  G.  Clinkscales.. 

Professor  of  mathematics. 

MATHEMATICAL   TEACHING   AT   THE   PRESENT   TIME.         301 
(a)  Universities  and  Colleges— Continued. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

147 

H.  H.Wright 

and  instructor  in  vocal 

music. 

148 

University  of  Tennessee. . 

Knoxville,  Tenn 

Wm,  W.Carson... 

Professor  of  mathematics 
and  civil  engineering. 

149 

Grant  Memorial  Univer- 
sity. 

Athens,  Tenn 

D.  A.  Bolton 

Professor  of  mathematics. 

150 

Bethel  College 

McKenzie,  Tenn 

Chattanooga,  Tenn. . 

W.  W.  Hamilton. . 

Do. 

151 

Chattanooga  University  . . 

E.  A.  Eobertson.. . 

Do. 

152 

Vanderbildt  University  . . 

N  ashville,  Tenn 

Wm.  J.  Vaughn . . . 

Do, 

153 

University  of  Texas 

Austin,  Tex 

G.B.Halsted 

Professor  of  pure  and  ap- 

plied mathematics. 

154 

Agricultural  and  Mechan- 
ical College  of  Texas. 

College  Station,  Tex. 

L.  L.  M'Innis 

Professor  of  mathematics. 

155 

Kandolph-Macon  College. . 

Ashland,  Va 

E.B.  Smithey 

Do. 

156" 

Emory  and  Henry  College. 
Hampden-Sidney  College.. 

S.M.Barton 

Do. 

167 

Hampden  Sidney,  Va 

J.R.Thornton.... 

Do. 

158 

University  of  Virginia 

Charlottesville,  Va. . 

C.  S.VenaWe 

Do. 

153 

Bethel  Military  Academy . 

Bethel  Academy  P. 
0.,  Va. 

E.S.  Smith 

Instructor  in  highermathe- 
matics  and  modern  lan- 
guages. 

160 

Virginia  Agricultural  and 
Mechanical  College. 

Blacksburg,  Va 

J.  E.  Christian 

Professor  of  mathematics 
and  civil  engineering. 

161 

Polytechnic  Institute 

Ife-w  Market,  Va 

W.H.Smith 

President. 

162 

Vermont  Methodist  Sem- 
inary. 

Montpelier,  Vt 

E.  A.  Bishop 

Principal. 

163 

Norvrich  University 

^orthfield.Vt 

J.B.Johnson 

Professor  of  mathematics. 

164 

University  of  Washington. 

Seattle,  "Wash 

J.M.Taylor 

Do. 

165 

Wheeling  Female  College. 

Wheeling,  "W.Va... 

H.  K.  Blaisdell 

President. 

166 

"West  Virginia  College 

Flemington,  W.  Va  . 

T.E.Peden 

Do. 

167 

C.  H.  Chandler 

Professor  of  mathematics 

and  physics. 

168 

Beloit,  Wis 

T.A.  Smith 

Do. 

Are  students  entering  your  institution  thorough  in  the  mathematiGS  required  for  admission  f 

"No,"  "not  generally,"  "by  no  means,"  "seldom:"  3,5,6,7,8,9,12,13,14,16,17, 
19,20,21,22,23,25,27  (but  growing  better),  29,31,33,36,39.40,42,43,44,47,49,51, 
52, 53, 56, 57, 58, 59, 60,  61,  63,  66, 67,  68, 69, 70, 72, 73, 76,  80,  82,  83, 86,  88, 91,  92, 93  (but 
marked  improvement  every  year),  94, 96, 97, 99, 102, 104, 105, 109, 110,  111,  113, 117, 160, 
161, 162, 163, 164, 165, 166, 167, 168  (and  this  is  one  of  the  evils  of  our  times). 

"  Fairly  so,"  "  reasonably  so : "  10, 11, 15, 30. 32, 34, 37, 46, 55, 84, 95. 

"Not  as  thorough  as  we  desire :  "   35, 38, 41, 71, 74, 87, 107, 108, 114, 116. 

"  Yes : "  1  (most  of  them),  28, 45  (generally),  54  (or  fail  to  enter),  64, 65  (usually),  78 
(a  fair  proportion),  79  (generally),  89,  90  (or  fail  to  enter),  100  (or  they  are  placed  in 
preparatory  department),  101, 112  (generally),  115. 
Is  the  mathematical  teaching  hy  text-looh  or  iy  lecture? 

This  question  was  answered  by  all  who  sent  in  reports.  The  following  forty-eix 
answers  were  "by  text-book,"  without  indicating  that  any  lectures  whatever  were 
given:  6,7,11,12,14,17,18,20,21,24,25,26,29,30  (it  is  impossible  to  teach  mathe- 
matics by  lecture),  32,  33,  37, 44, 50, 51, 53,  63, 65, 68,  73, 74,  85, 88, 101, 10.^,  105, 107, 109, 
113, 114, 125, 138, 141, 142, 147, 149, 150, 151, 161, 163, 164. 


302  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

The  following  sixty-five  answers  were  "  mainly  by  text-book,"  "  text-book  princi- 
pally," "text-book  as  a  basis,"  "text-book  and  informal  lectures,"  or  some  similar 
phrase,  iudicating  that  the  text-book  predominated :  1,  3,  8,  9, 15, 19, 22, 28, 31, 34, 38, 
40,  41,  43,  45,  47, 48, 52, 55,  57, 58,  59, 60,  62, 66, 71,  72,  75, 77, 80,  86,  87, 89, 90,  91, 92, 93, 95, 
97,  100, 103, 106, 110, 116, 118, 119, 120, 122, 123, 128,  133,  134,  135,  136,  137,  143,  144,  148, 
155, 156, 159, 162, 166, 167, 168. 

The  following  fifty-five  answered  "by  both,"  or  "by  text-book  and  lecture,"  with- 
out saying  which  predominated  :  2, 4,  5, 10, 16,  23, 27,  35,  36, 39,  42, 46, 49,  54, 56, 61, 64, 
67,69,70,76,78,79,82,83,84,94,96,98,99,104,  108,  111,  112,  115,  117,  121,  124,  126,  127, 
129, 130, 131, 132, 139, 140, 145, 146, 152, 153, 154, 157, 158, 160,  165.  The  answer  of  num- 
ber 13  is  "  by  lectures,  except  elementary  geometry ;  "  and  of  81,  "  by  lecture." 
What  mathematical  journals  are  taken? 

The  following  answered  "  none  :  "  1, 3, 5, 6, 7, 9, 14, 15, 17, 20, 21, 22, 24, 25, 29, 30, 32, 
33, 36, 39, 40, 43, 44, 45, 48, 49, 50, 53, 56,  57,  58,  59,  60,  63,  65,  69, 70,  71,  73, 74, 76, 77, 78, 79, 
80, 81, 82, 85, 87, 88, 89, 91, 94,  96,  99, 101, 102, 103, 104, 105, 107, 108, 112,  114, 116,  117,  121, 
122, 128,  129,  133,  134, 141,  147,  149,  150,  151,  159,  160,  162.  163,  165,  167,  168.  Some  of 
these  answers  were  "none  by  the  college,"  "none  that  are  purely  mathematical," 
"several  scientific  and  engineering  j  ournals, "  but  most  of  them  were  simply  "  none." 
In  addition  to  this  list,  numbering  eighty-four,  we  may  safely  add  thirty-three  that 
did  not  answer  this  question  in  their  report,  making  one  hundred  and  seventeen  in- 
stitutions out  of  one  hundred  and  sixty  eight  that  take  no  mathematical  journal  of 
any  sort  devoted  to  pure  mathematics. 

The  following  reported  as  taking  simply  the  American  Journal  of  Mathematics: 
10, 27, 28, 54, 55, 72,  75, 84, 95, 131, 145. 

The  following,  as  taking  simply  the  Annals  of  Mathematics :  4, 8, 13, 16, 23, 42, 64, 
90, 100, 110, 148. 157. 

University  numbered  11  is  taking  d,  X;,  Z,  marked  below;*  12,i, d,n;  19,  fc;  35,1),  d, 
i,s;  37,  A;;  38,b,d,q,n ;  il,d,m,l;  46,&,fc,eto.;  47,&,d;  51,6, nearly  all  the  foreign 
journals  are  expected  after  this  year ;  67,  all  the  leading  ones ;  68,  nearly  all  mathemat- 
ical journals;  83,  b,  d,  j  ;  86,  h,  A,  u ;  92,  Ic,  I,  o,  etc. ;  93,  l,  d ;  97,  &,  j,  etc. ;  98,  h,  d,  n ; 
106,  a,  h,  Jahrbuch  d.  Vortsch.  d.  Math. ;  109,  d,  j,p  ;  111,  any  we  can  get ;  115,  a,  h,  c, 
d,e,f,g,h,i,j,m,n,p,a,t,u  ;  152,  b,d,j,m,p,s,t;  15'3,b,d,t;  154, d,s;  155, d,p;  158, d, 
jj,  and  others ;  161,  q  ;  164,  d,  h,  I. 

Are  there  any  mathematical  seminaries  or  clubs,  and  how  are  they  conducted  ? 
All  answered  in  the  negative,  except  the  following : 

15.  No  clubs,  unless  special  classes  for  voluntary  and  outside  reading  be  so 
designated.    Such  classes  are  conducted  like  all  other  classes. 

38.  A  club.  The  meetings  of  the  club  occur  on  alternate  Tuesdays.  Member- 
ship about  25  ;  topics  are  assigned  to  or  chosen  by  the  student  at  his  option ; 
assistance  is  given  him  as  he  may  need.  The  work  is  pedagogical,  rather  than 
original. 

41.  One.  Reading  and  exposition  of  the  more  difficult  parts  of  Williamson's 
Calculus. 

51.  In  connection  with  the  Science  Club  ;  by  lectures. 

67.  There  is  a  mathematical  society,  in  which  there  is  free  choice  of  subjects 
for  communication,  and  there  are  two  or  three  seminaries  for  post-graduate 
students,  conducted  by  the  teacher  on  special  lines. 

*(a)  Acta  Mathematica.  (&)  American  Journal  of  Mathematics,  (c)  Annali  di  Matematica.  (d) 
Annals  of  Mathematics,  (e)  Archiv  d.  Math.  u.  Physik.  (/)  Bulletin  des  Sciences  Math,  et  Astron. 
(g)  Bulletin  de  Soci6t6  Math,  (h)  Comptes  Eendns.  (i)  Journal  de  Math.  (Lionrille).  (j)  Journal 
f.  reine  u.  angew.  Math.  (Crelle).  (k)  Mathematical  Magazine.  (I)  Mathematical  Visitor,  (nt)  Mathe- 
matische  Annalen.  (n)  Messenger  of  Mathematics,  (o)  Proceedings  London  Math.  Society,  (p) 
Nouvellea  Annales  de  Math,  (q)  School  Visitor,  (r)  School  Messenger,  (s)  Quarterly  Journal  of 
Math,    (t)  Zeitachrif t  f.  Math.  u.  Phys.    (u)  Zeitschrift  {,  Vermessungskunde. 


MATHEMATICAL   TEACHING   AT   THE   PRESENT   TIME.         303 

Are  there  any  mathematical  seminaries  or  clubs,  and  hoiv  are  they  conducted  ? — Continued. 

115.  Yes,  one.  It  proposes  to  discuss  the  literature  of  mathematics,  to  solve 
problems  given  by  members,  and  to  make  original  investigations. 

136.  No  clubs,  but  seminaries,  through  part  of  regular  course,  but  not  very 
formal ;  they  are  intended  to  afford  students  opportunity  of  working  problems 
under  guidance. 

143.  There  are  men  in  each  class  studying  for  honors.  The  principal  part  of 
their  work  is  the  solution  of  original  problems.  I  meet  them  frequently  for 
discussions  and  suggestions. 

158.  No  clubs;  the  lectures  and  recitations  regularly  extend  through  an /i  our 
and  a  half,  and  at  each  original  solutions  of  problems  are  given,  and  next  time 
are  called  for.    Each  meeting  of  each  class  is  a  seminarium. 

Are  there  any  scholarships  or  fellowships  for  graduate  students  in  matheinatics  f 
All  who  answered  this  said  "  no,"  except  the  following  : 

67.  Yes. 

84.  None  for  mathematics  exclusively. 

86.  One  is  occasionally  given  to  a  man  of  high  promise. 

93.  Fellows  are  allowed  to  choose  mathematics. 

97.  One  fellowship ;  two  scholarships  await  instantly  State  appropriation 
for  support. 

115.  Yes,  an  annual  fellowship  in  science. 

136.  One  fellowship  awarded  yearly  to  a  properly  qualified  graduate  of  any 
college. 

145.  Yes,  there  is  one,  retainable  for  two  years. 

159,  Occasionally  conferred  on  deserving  students  wishing  to  prosecute  their 
studies  at  other  institutions. 

Is  the  percentage  of  students  electing  higher  mathematics  increasing  or  decreasing  ? 

The  following  reported  "increasing:"  1,  5,  6,8,9, 12, 15  (among  scientific  students), 
16, 21, 24, 25, 26, 27,  28,  31,  32, 33  (?),  35, 37,  38, 39, 40, 42  (?),  46, 50, 51, 53, 54, 55, 56  (with 
gentlemen),  57,  59,  63,73,75,76,78,79,82,88,  90,93,97,98,  99,  101,  102,  106,108,111, 
112, 113,  116,  117  (?),  119, 123, 124, 125, 126, 128, 129, 132, 137, 140, 142, 144, 145, 149, 150, 
151, 153, 154, 156, 157, 158, 159, 160, 161, 164,  165,  166. 

The  following  reported  "about  constant,"  " neither  increasing  nor  decreasing : " 
3, 4, 23, 30, 48,  67, 70, 74, 80, 84,  86, 91, 92, 100, 104, 107, 114, 115,118, 122, 127, 130, 131, 141, 
147, 148, 162, 163. 

The  following  reports  indicated  a  "  decrease :"  15  (among  classic  students),  44, 49, 
56  (with  ladies),  83. 

The  following  reported  that  none  of  the  mathematical  studies  were  elective:  10, 
11, 17, 18, 20, 22,  36,  41, 45,  47, 52, 65, 71, 85, 87, 94, 96, 103, 105, 108, 120, 133, 135, 139, 142, 
146,155. 

Does  the  interest  in  mathematics  increase  as  students  advance  to  higlier  subjects  ? 

"Yes:"  3,5,6,8,10,11,13,16,17,18,21,22,23,26,28,29,  30,  32,  35,  36,  37,  38,  40,  41, 
42,46,48,50,51,52,53  (very  much),  54  (generally),  55, 57, 59, 60, 61,  63,65,66  (?),73,77, 
79,82,84,87  (generally),  88, 90, 92, 93,  94,  95, 98, 101,  102,  104  (?),  105  (?),  108,  111,  112, 
113,116,118,119,122,123,126,131,132,  133,  134,  139,  140,  145  (generally),  146,  147,150, 
151, 153, 154, 156, 157, 158  (emphatically  so),  159, 160, 161, 163, 164, 165, 166, 167. 

"With  the  best  students  only,"  "with  those  students  whose  mental  tendencies  are 
along  mathematical  lines,"  or  some  similar  remark :  1, 4, 9, 12, 24, 25,  31, 33,  34,  43,  44, 
45, 47, 56, 58, 68, 70,  71,72, 74, 75,  76,  80,  81,  85, 107, 114, 120, 121, 124, 125, 127, 128, 129, 130, 
136, 137, 141, 142, 143, 148, 149, 152, 155. 

"The  interest  increases  so  long  as  the  student  sees  the  bearing  of  his  work  upon 
practical  scientific  investigation  or  can  be  assured  that  it  has  such  a  bearing."  "It 
increases  as  application  to  practical  matters  is  shown : "  15, 19, 69, 103. 


304  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

"All  who  understand  the  principles  show  a  growing  interest,"  "where  proper 
preparation  on  the  part  of  the  student  has  been  attended  to  and  the  teacher  is  a  live 
man,  it  does"  (110),  "  The  interest  is  according  to  the  clearness  of  apprehension  of 
mathematical  truths.  Hence,  the  more  evolved  or  abstruse  the  matter,  the  greater 
the  interest  to  those  who  succeed  "  (144) :  27, 91, 97, 110, 115, 144. 

"No  : "  2, 7, 20,  34  (for  poor  students),  39, 43  (for  the  majority),  49, 86, 100, 106, 109, 
135, 162, 168. 

Are  prizes  awarded  for  excellence  of  daily  class-room  worh,  or  for  success  in  orginal 

research  f 

"  No  prizes  awarded : "  1, 4, 6, 7, 8, 10, 13, 14, 16, 17, 19, 20, 21, 23, 28, 29,  30, 31, 32, 33, 
34, 35, 37,  38, 39, 40, 41, 43, 46, 47,  48,  49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65,  68, 69, 
70, 71, 72,  73, 76, 78,  79, 80, 81,  84, 85,  87, 88, 89, 90, 91, 93, 94, 95, 98, 100, 101, 103, 104, 105, 
106, 107, 108,  111,  112, 114, 115  (except  class  honors),  119, 120, 121, 122, 123, 124, 125, 126, 
127, 128, 129,  131, 132, 133,  135, 136, 138,  139, 140, 141, 142, 144, 145, 147, 148, 149, 150, 151, 
152, 154, 156, 157, 158, 160, 161, 164, 166, 167, 168. 

"For  both:"  2,  9,36,67  (bestowal  of  scholarships  and  fellowships  is  based  upon 
both  the  considerations),  82,  99, 134, 155  (but  I  do  not  believe  in  prizes). 

"  For  work  in  the  class-room :  "  18, 22, 25, 26, 62, 66, 77, 86, 96, 118, 137, 143, 159, 162. 

"  For  outside  work,  not  generally  original : "  15, 116. 

"  For  original  investigation  only  :  "  97, 102, 117. 

"  Yes,  prizes  are  awarded : "  3,5,11,12,24,27,54,57,74  (scholarship  of  $300  to  best 
Sophomore  in  mathematics),  92, 97, 109, 110, 113, 165. 

What  mathematical  subjects  are  preferred  iy  students? 

The  answers  given  point  to  no  definite  conclusion.  For  want  of  space,  they  are 
here  omitted,  except  the  following:  "  Their  preferences  are  generally  for  the  particular 
subject  which  they  have  had  the  best  elementary  training  in  "  (148). 

Are  topics  assigned  to  students  for  special  investigation  ? 

1.  Yes.  ? 

2.  Problems  are  proposed. 

3.  Sometimes  prize  problems  are  given  to  students, 

5.  Yes. 

6.  Yes. 

7.  For  the  higher  classes. 

8.  Yes;  often. 

9.  Yes. 

10.  Not  in  general. 

11.  To  a  small  extent. 

12.  Occasionally. 
14.  No. 

17.  Yes. 

18.  Sometimes. 

19.  Occasionally. 

20.  No. 

21.  Yes. 

22.  No. 

23.  Yes. 

24.  Yes. 

25.  No. 

26.  There  are. 

27.  Independent  problems  giren  in  all  the  classes  for  solution,  reported  OD 
paper. 

28.  Once  each  term. 

29.  No. 


MATHEMATICAL   TEACHING  AT   THE   PRESENT   TIME.        305 

Are  topics  assigned  to  students  for  special  investigation  t — Continued. 

30.  Not  much.    The  man  who  pursues  original  investigation  with  the  aver- 
age student  will  make  a  failure. 

31.  We  have  not  been  accustomed  to  do  so. 

32.  Not  to  any  extent  that  deserves  mention. 

33.  Practically,  no. 

34.  Not  to  under-graduates.    There  are  no  graduate  students  in  mathematics 
at  present. 

35.  Yes. 

36.  Yes ;  hut  necessarily  elementary. 

37.  Yes,  sir. 

38.  Yes,  in  connection  with  the  club  and  for  graduation  theses. 

39.  Yes. 

40.  Sometimes ;  but  our  classes  are  generally  so  closely  occupied  by  their 
various  studies,  there  is  but  little  time  for  extra  work. 

41.  No. 

42.  Yes. 

43.  Yes. 

44.  Frequently. 

45.  They  are,  and  form  a  very  essential  part  of  the  worlc. 

46.  Students  in  the  higher  classes  are  assigned  such  topics. 
^              47.  None  advanced  enough. 

48.  Yes. 

49.  Sometimes. 

51.  Yes. 

52.  Occasionally. 

53.  Yes. 

54.  Yes. 

55.  No. 

56.  Yes. 

58.  Yes. 

59.  Yes ;  with  good  saeeees. 

60.  Yes. 

63.  Yes. 

64.  No. 

65.  No. 

67.  They  are  in  the  seminaries. 

68.  No. 

69.  A  few. 

70.  To  a  limited  extent. 

71.  No. 

72.  Not  to  any  great  extent. 

73.  Occasionally. 

74.  In  elective  divisions. 

76.  To  some  extent. 

77.  Rarely. 

78.  Yes. 

79.  No. 

81.  Yes. 

82.  Yes. 

84.  No. 

85.  No. 

86.  Yes,  to  post-graduates. 

87.  Yes. 

88.  Not  often. 

881— No.  3 20 


306  TEACHING  AND   HISTORY   OF  MATHEMATICS. 

Are  topics  assigned  to  students  for  special  investigation? — Continued. 

89.  In  applied  mechanics,  yes. 

90.  Not  for  original  investigation,  but  otherwise. 

91.  Occasionally. 

92.  Yes,  for  thesis  in  general  geometry  and  calculus. 

93.  Yes,  especially  in  elementary  geometry,  analytical  geometry,  and  calculus. 

94.  Yes. 

95.  Eare. 

96.  Only  to  a  limited  extent. 

97.  Such  assignment  has  hitherto  been  only  exceptional,  hereafter  to  be  made 
regular. 

98.  In  applied  mathematics  theses  are  required  on  special  subjects,  and  origi- 
nal investigation  encouraged. 

99.  Yes. 

100.  No. 

101.  Yes. 

102.  Yes,  in  all  the  different  branches,  especially  in  applied  mathematics. 

103.  The  graduating  and  other  theses  are  on  subjects  divided  among  the 
departments. 

104.  No. 

105.  No. 

106.  They  are. 

107.  Yes. 

108.  Yes. 

109.  No. 

110.  Yes,  this  is  encouraged  in  all  the  classes,  but  is  secured  best  in  the 
higher  classes. 

111.  Yes. 

112.  Occasionally  original  theorems  and  problems  are  given. 

114.  Occasionally,  results  submitted  in  original  theses. 

115.  No,  except  the  work  done  in  the  seminary. 

116.  Yes,  in  all  classes  of  all  departments. 

117.  No. 

118.  Yes,  to  a  large  extent  in  geometry. 

119.  In  pure  mathematics  very  seldom;  not  in  applied  mathematica, 

120.  Yes. 

121.  No. 

122.  No. 

123.  No.' 

124.  We  have  this  in  view  for  next  term. 

125.  None  as  yet. 

126.  Yes. 

127.  Yes. 

128.  Yes. 

129.  To  some  extent. 

130.  Yes. 

131.  Occasionally. 

132.  No. 

133.  To  some  extent. 

134.  Occasionally. 

135.  No. 

136.  I  should  consider  this  exercise  profitable  only  to  very  advanced  students ; 
and  have  not  had  occasion  to  employ  it  yet. 

137.  Yes,  to  some  extent, 

138.  No. 


MATHEMATICAL    TEACHING  AT   THE   PRESENT   TIME.         307 

Are  topics  assigned  to  students  for  special  investigation? — Continued 

139.  No. 

140.  Occasionally. 

141.  Not  to  any  extent. 

142.  Yes,  in  applied  geometry,  surveying,  and  pbysics. 

143.  Occasionally  to  advanced  students. 

144.  Only  exercises  in  theorems  and  problems. 

145.  Yes. 

146.  Yes, 

147.  No. 

148.  The  solution  of  problems  related  to  the  recitations  is  reiiuired.  Nothing 
else. 

149.  No. 

150.  No. 

152.  No. 

153.  Yes. 

154.  Yes. 

155.  In  theNhigher  classes  topics  are  occasionally  assigned. 
1/56.  No. 

157.  Original  exercises  are  given  at  intervals. 

158.  To  graduate  students,  candidates  for  the  degree  of  Ph.  D. 

159.  But  few  outside  of  text-book, 

160.  Yes. 

161.  No. 

162.  No, 

163.  No. 

164.  Yes. 

165.  Yes. 
.    166.  Yes. 

167.  To  some  extent. 

168.  Occasionally.* 

(o)  Is  any  attention  given  to  the  history  of  mathematics  f      (5)  Does  it  malce  the  suhject 

more  interesting  ? 

(a)  "  Yes  :  "  1, 5, 9, 18, 34, 35,  37,  39, 42,  44, 46, 48, 52,  53,  61,  63, 64,  65,  72, 80,  81,  90,  92, 
97, 98, 99, 101, 102, 108, 112, 114, 116, 123, 126, 129, 131, 135, 136,  138,  145, 152, 153  (great), 
154, 156, 157, 158, 160, 164. 

(rt)  "  Very  little,"  "only  incidentally,"  " not  much",  etc.:  4, 6, 8, 11, 12, 13, 16, 17, 19, 
21,  23, 24, 25, 27, 28, 30, 38, 40, 41,  43, 45,  47,  51, 54, 55,  56, 58, 59, 60, 66, 67, 68, 73, 74, 75, 76, 
78,79,82,83,88,91,93,94,100,104,106,107.111,115,118,119,1203  121,  124,  125,  127,128, 
130, 133, 137, 143, 144, 147, 151, 159, 163, 168. 

(o )  "  No  :"  3,  7, 10, 14, 20, 22, 29,  32, 33, 36, 49,  50, 57,  69,  70,  71 ,  77,  83,  85,  87,  89,  95,  96, 
103, 105, 109, 110, 117, 122, 132, 134, 139, 140, 142, 146, 148, 149, 150, 161, 162, 166, 167. 

(6)  "Yes,"  "it  does,"  "most  decidedly,"  was  the  experience  of  all  who  had  given  any 
attention,  in  the  class-room,  to  mathematical  history,  except  the  following,  who  were 
in  doubt :  11, 15, 16, 47, 56, 76, 104, 120.  Even  these  were  inclined  to  say  "  yes."  Xo  one 
answered  that  it  did  not  make  the  subject  more  interesting — a  clear  case. 

How  does  analytical  mathematics  compare  in  disciplinary  value  tvith  synthetical  f 

1.  I  regard  both  methods  equally  important. 

4.  I  think  synthetical  has  much  the  greater  disciplinary  value ;  analytical 
has  much  the  greater  value  for  practical  application.  Analysis  is  the  princi- 
pal tool  for  investigation  and  work. 

*  Widely  difforont  vie-svs  seem  to  be  implied  in  the  above  answers  aa  to  what  constitutes  a  "topio 
for  special  investigation," 


308  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

How  does  analytical  mathematics  compare  in  disciplinary  value  with  synthetical? — Cont'd, 

5.  Analytical  superior. 

6.  The  former  uaed  more  largely  in  the  Grammar  Department. 

8.  Analytical  mathematics  gives  the  better  mental  discipline. 

9.  I  think  both  necessary  to  full  mental  development,  but  if  I  were  obliged 
to  choose  I  should  prefer  analytical. 

10.  I  can  not  say  fairly,  for  my  teaching  has  been  wholly  in  analytical  mathe- 
matics.   In  my  studies  I  prefer  that  method. 

11.  I  use  combination  of  both  and  so  can  not  well  answer. 

12.  The  development  of  reason  is  more  regular,  rapid,  and  substantial  in 
geometry  than  in  any  other  branch  of  the  mathematical  course.  For  advanced 
students  I  would  count  algebraic  analysis  a  superior  discipline. 

13.  It  seems  to  be  a  question  of  individuality. 

15.  I  regard  analytical  mathematics  as  the  more  valuable  and  the  more  im- 
portant. 

16.  The  former  is  superior. 

17.  It  is  superior. 

18.  Analytical  seems  to  be  better. 

19.  Common  geometry,  considered  as  an  application  of  logic,  especially  in 
the  demonstration  of  easy  "riders"  and  in  very  simple  exercises  in  construc- 
tions, is  of  pre-eminent  value  to  quite  young  and  undisciplined  minds.  At  dif- 
ferent stages  each  has  its  peculiar  and  really  unmeasurable  value. 

20.  They  are  of  equal  value. 

21.  I  have  not  data  enough  for  an  opinion. 

22.  Superior,  if  the  two  are  divorced;  but  the  synthetical  should  be  united 
•with  it. 

23.  Favorably. 

24.  Analytical  greater. 

25.  With  the  majority  of  students  more  satisfactory  results  are  obtained 
through  the  synthetical  method  of  reasoning. 

26.  Analytical  preferred. 

27.  As  a  rule,  I  have  found  that  students  stand  better  in  geometry  than  in 
algebra.  When  analytical  geometry  is  clearly  comprehended,  it  affords  the  best 
discipline  for  the  mind. 

28.  Synthesis  seems  to  give  better  discipline. 

30.  Analytical  preferable. 

31.  The  former  is  a  better  test  for  form  and  figure,  the  latter  seems  to  tax 
the  memory. 

32.  We  have  no  classes  sufiQciently  advanced  to  test  the  relative  value  ex- 
tensively. 

33.  If  the  work  be  the  same  in  both,  the  synthetical. 

34.  My  own  preference  is  analytical. 

35.  Could  not  get  along  with  either  method  left  out  (Professor  Shattuck). 
Each  has  its  special  function  ;  as  well  ask  whether  braces  or  tie-rods  are  of 
most  service  in  a  bridge-truss  (Regent  Peabody). 

36.  Synthetical  more  valuable. 

37.  Disciplinary  value  of  former  is  greater. 

38.  We  give  the  analytical  the  first  consideration  after  the  student  is  led  up 
to  it. 

40.  Superior. 

41.  I  think  the  former  the  more  valuable  as  an  instrument  of  research,  the 
latter  as  a  means  of  discipline. 

42.  The  analytical  is  more  valuable  simply  as  a  means  of  discipline. 

43.  The  synthetical  is  better  for  younger  students ;  the  analytical  for  those 
more  mature. 


MATHEMATICAL   TEACHING   AT   THE   PEESENT   TIME.        309 

Hov3  does  analytical  mathematics  compare  in  disciplinary  value  tvith  synthetical? — Cont'd, 

45.  It  is  evidently  far  superior. 

46.  Each  affords  excellent  discipline. 

47.  It  is  superior. 

48.  For  college  grade  we  think  the  analytical  produces  the  best  results. 

49.  Better. 

50.  Better  for  discipline,  but  we  have  not  used  it  as  yet. 
nl.  Somewhat  superior  in  value. 

52.  Superior. 

53.  Analytical  training  is  more  beneficial. 

54.  Favorably. 

55.  They  are  superior. 

56.  Prefer  the  analytical. 

57.  We  use  both  methods,  but  give  preference  to  former. 

58.  Can  we  do  without  either  ?  I  should  say  both  are  necessary,  but  analyt- 
ical is  less  taught. 

59.  Analyzing  the  whole  into  its  elements  is  valuable,  but  building  the  whole 
from  elements  is  very  valuable. 

60.  Superior  for  advanced  students. 
63.  The  analytical  the  more  valuable. 

65.  Analytic  mathematics  is  far  superior  in  its  disciplinary  value. 

67.  The  latter  is  probably  the  more  valuable  discipline  in  early  stages  of  a 
mathematical  education ;  but  after  the  elements  of  geometry  are  mastered, 
probably  the  reverse  is  true. 

68.  In  general  we  prefer  analytical  methods. 

69.  Latter  preferred, 

70.  Doubtful ;  students  prefer  synthetical. 

73.  In  my  judgment  the  analytical  method  is  to  be  preferred. 

75.  For  the  average  student  the  synthetical  gives  better  results. 

76.  I  think  analytical  mathematics  better  for  mental  discipline. 

79.  Its  disciplinary  value  is  less  than  that  of  the  synthetical. 

80.  The  synthetical  is  more  valuable. 

81.  They  interact;  but  the  latter  is  an  indispensable  prerequisite  for  the 
former. 

82.  I  should  place  analytical  as  greater  in  disciplinary  value. 
84.  Analytical  is  inferior  to  synthetic, 

86.  Both  methods  are  essential  and  I  am  not  aware  of  any  difference.  Per- 
haps I  do  not  understand  the  question. 

87.  Superior;  yet  this  depends,  perhaps,  on  the  mind  of  the  individual 
student. 

90.  Very  favorably,  so  far  as  our  experience  has  gone. 

91.  I  prefer  the  former  for  advanced  students— the  latter  for  beginners,  or 
students  of  a  low  grade. 

92.  Synthetical  seems  best  for  the  less  advanced  students. 

93.  I  do  not  believe  that  the  two  can  be  separated  and  compared.  I  believe 
with  Sir  William  Hamilton,  "Analysis  and  synthesis  are  only  the  two  necessary 
parts  of  the  same  method.  Each  is  the  relative  and  correlative  of  the  other." 
Neither  without  the  other  would  be  of  much  value. 

94.  The  synthetical  is  absolutely  necessary  as  a  foundation  of  good  work; 
after  the  foundation,  tbe  former  is  desirable. 

95.  Do  you  mean  graphical  (or  geometrical)  by  synthetical  ?  I  think  de- 
scriptive geometry  has  the  finest  disciplinary  value. 

97.  As  commonly  taught,  unfavorably  ;  as  taught  here,  with  special  stress  OU 
Morphology  and  by  aid  of  determinants,  very  favorably. 

98.  In  favor  of  the  former. 


310  TEACHING    AND   HISTOEY   OF  MATHEMATICS. 

How  does  analytical  mathematics  compare  in  disciplinary  value  with  st/nthetical  f — Cont'd. 
99.  Are  in  favor  of  the  analytical. 

101.  Superior. 

102.  That  depends  upon  the  peculiar  natural  bent  of  the  pupil's  mind.  For 
some,  analysis,  and  for  others,  synthesis  is  more  valuable. 

104.  Can  not  institute  comparison,  we  use  both  in  combination. 

106.  I  would  answer  this  by  saying,  that  I  consider  special  geometry  better 
for  mental  discipline  than  analytical  geometry,  and  geometry  better  than 
algebra. 

109.  Analytical  superior, 

110.  In  my  judgment  the  analytical  is  so  far  superior  to  the  synthetical  that 
there  is  left  little  room  for  comparison.  Permit  me  to  say  that  reason  wants 
light,  not  darkness. 

111.  They  are  about  equal.    We  use  Peck's  and  Davies'  methods. 

112.  The  comparison  is  in  favor  of  analytical  mathematics. 

113.  The  analytical  method,  in  my  opinion,  produces  better  results  than  the 
synthetical. 

114.  Superior. 

115.  Each  has  its  special  value ;  both  are  desirable  (Professor  Peck).  An- 
alytical gives  the  more  rigorous  training.  Each  plays  its  own  part  (Tutor 
Fiske). 

116.  Synthetical  better  for  training  in  formal  logic ;  in  other  respects  analyt- 
ical is  unquestionably  superior. 

117.  Synthetical  seems  to  give  better  results. 

118.  For  older  students  the  analytical  methods  are  superior;  for  those  below 
the  Sophomore  class,  this  is  doubtful. 

119.  I  think  the  analytical  is  better. 

120.  Analytical  mathematics  predominates  here,  and  therefore  has  the  greater 
disciplinary  value.  If  comparing  equal  times  in  the  two,  I  should  say  syn- 
thetical. 

122.  Synthetic  best  for  discipline  ;  analytic  best  for  use. 

123.  Former  is  better. 

125.  It  is  hard  for  me  to  answer  this.  Perhaps  the  latter  is  superior  for  dull 
or  average  students,  while  the  former  is  preferable  for  the  more  able. 

126.  Both  necessary  for  proper  discipline. 

128.  Well. 

129.  Analytical  mathematics  is  the  better.  / 

130.  Sometimes  seems  to  me  superior;  sometimes  seems  to  me  inferior,  de- 
pending upon  the  mental  character  of  the  student. 

132.  We  teach  no  synthetical  mathematics  in  the  university,  except  a  book 
of  elementary  mechanics,  which  is  good  in  its  place,  but  analytical  mathe- 
matics alone  develops  real  mathematical  power. 

133.  I  regard  the  analytic  method  as  much  superior  in  way  of  developing 
habit  and  power  of  investigation. 

134.  I  use  both  and  would  not  willingly  part  with  either.  Deem  them  of 
about  equal  value. 

135.  Equally  valuable  though  in  diflferent  way. 

136.  I  should  be  inclined  to  give  preference  to  analytical;  but  where  there  is 
a  strong  natural  mathematical  bent,  possibly  more  discipline  is  derived  from 
synthetical  mathematics. 

137.  Rather  unfavorably  with  the  average  student. 

139.  Superior. 

140.  Better. 

143.  Both  valuable ;  both  necessary, 

144.  Analytical  is  favorable  for  advanced  students ;  synthetical,  for  younger 
students. 


MATHEMATICAL  TEACHING  AT  THE  PEESENT  TIME.   311 

Hotodoes  analytical  mathematics  compare  in  disciplinary  value  ^vith  synthetical  f — Cont'd. 

145.  The  value  of  the  discipline  depends  upon  the  closeness  of  the  student's 
application  rather  than  upon  the  methods  employed. 

146.  Superior  to  it. 

147.  I  consider  the  analytical  far  superior  to  the  synthetical. 

148.  In  my  opinion,  the  analytical  is  far  superior  to  the  synthetical. 

149.  I  think  the  analytic  is  better. 

150.  The  former  is  of  more  disciplinary  value  than  the  latter. 

151.  The  analytical  mathematics  in  most  cases  most  satisfactorily  fulfils  the 
end  or  object  of  mathematical  study. 

152.  The  former,  in  my  opinion,  is  preferable  in  almost  every  respect. 

153.  Analytical  mathematics  is  very  far  inferior  to  synthetic  in  disciplinary 
value. 

155.  Analytical  mathematics  has,  I  think,  a  higher  disciplinary  value  than 
synthetical. 

156.  The  synthetical  is  more  valuable,  I  think,  but  by  no  means  should  either 
be  adopted  to  the  exclusion  of  the  other. 

158.  Impossible  to  make  a  comparison  in  so  short  space. 

159.  I  regard  analytical  mathematics  as  possessing  higher  disciplinpry  value, 
when  properly  taught. 

160.  Analytical  mathematics  is  superior  to  synthetical  in  disciplinary  value. 

161.  Favorably,  both  should  be  used. 

163.  I  favor  analytical  mathematics. 

164.  Analysis  is  superior  in  disciplinary  value. 

165.  Superior. 

166.  They  are  about  equal. 

167.  Any  true  method  of  study  seems  to  me  to  use  them  both,  with  so  frequent 
changes  that  comparison  is  difficult.  Moreover,  their  relative  value  differs 
with  different  pupils.* 

(o)  What  method  of  treating  the  calculus  do  you  favor,  that  of  limits,  the  infinitesimal,  or 
some  other?  (b)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student  ? 

I.  (a)  Limits.    (6)  Does  not  satisfy  the  student. 

3.  Calculus  is  not  taught  in  this  college. 

4.  (a)  I  favor  the  method  of  rates,  though  I  use  the  method  of  limits  and 
infinitesimals — the  latter  in  mechanics.     (6)  It  does  not  in  my  experience. 

5.  We  do  not  teach  it. 

8.  (a)  That  of  limits.  (&)  At  first  it  does  not  seem  rigorous  to  the  student, 
nor  to  satisfy  his  mind. 

9.  (a)  We  think,  with  many  others,  the  subject  needs  both,  (b)  Not  suffi- 
ciently so,  and  hence  the  advantage  of  calling  "limits"  to  its  aid. 

10.  (a)  Calculus  is  not  taught  here.  Personally  I  prefer  infinitesimals.  (&)  I 
think  so,  more  than  that  of  limits,  which  is  better  for  the  mathematician  than 
the  student. 

II.  (a)  I  favor  none  exclusively.  I  teach  "rates",  "infinitesimals",  and 
"limits."    (&)  It  does  not  seem  rigorous. 

12.  (a)  The  method  of  limits  (now  made  familiar  in  geometry)  seems  most 
satisfactory.     (&)  Not  to  beginners.   Later  this  method  should  be  studied  also. 

13.  (a)  Defining /' (a)  as  co-efficient  in  development : /(x)  =/ (a) -|-/'  (a) 

(x—a)  + (&)  It  does  not  seem  rigorous  as  usually  represented,  but 

could  be  made  so,  but  I  doubt  whether  for  beginners. 

15.  (a)  I  favor  the  method  known  as  that  of  "  rates."    (b)  I  think  not. 

♦For  collateral  reading  on  this  question  see  President  C.  "W.  Eliot's  article,  "What  ia  a  Liberal 
Education?"  in  the  Century  Magazine,  June,  1884;  Keport  of  the  (English)  Commission  in  1852; 
Report  of  the  French  Commissioners  in  1870. 


312  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

(o)  What  method  of  treating  the  calculus  do  you  favor,  that  of  limits,  the  infinitesimal,  or 
some  other?  (b)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student  ? — Continued. 

16.  (a)  A  combination  of  limits  and  infinitesimals.  (6)  Combined  with  the 
metliod  of  limits,  it  does;  alone,  no. 

17.  (a)  Infinitesimal ;  a  little  about  limits.  (&)  I  have  never  yet  had  a  stu- 
dent to  whom  I  could  not  make  it  perfectly  satisfactory. 

18.  (a)  Limits.     (6)  No. 

19.  (a)  In  theory,  Buckingham's  "  direct  method  of  rates;  "  practically,  the 
infinitesimal  as  set  forth  by  Olney  and  others,  on  account  of  its  practical  ad- 
vantages. (&)  The  philosophy  at  the  base  of  this  method  seems  to  involve  one 
in  a  maze  of  absurdities,  but  I  have  had  too  little  experience  with  pupils  in  the 
calculus  to  speak  positively  upon  this  point. 

20.  We  do  not  teach  the  calculus. 

21.  (a)  Doctrine  of  limits.     (6)  It  does  not. 

22.  (a)  That  of  limits.    (&)  Not  in  every  case. 

23.  (a)  Limits. 

24.  (a)  Limits. 

25.  (a)  The  method  of  limits,    (t)  It  does  not.  ' 

26.  (a)  Limits. 

27.  (a)  I  explain  and  illustrate  both  limits  and  infinitesimal  analysis,  (b) 
When  properly  explained  and  illustrated,  I  think  it  does. 

28.  (a)  The  infinitesimal,    (h)  It  does,  i.  e.,  generally. 

30.  Method  of  limits,  not  the  Newtonian  of  passing  to  zero. 

31.  (a)  Have  been  accustomed  to  take  the  limits. 

32.  We  have  no  classes  in  calculus. 

33.  (a)  Had  experience  only  with  infinitesimals.  (J)  No ;  certainly  Olney's 
presentation  can  be  improved  upon. 

34.  (a)  The  infinitesimal,  if  properly  presented.  (6)  Yes,  when  the  student 
can  appreciate  mathematical  reasoning. 

35.  (a)  Teach  both  methods,  do  not  favor  either,  {b)  Yea ;  Lagrange's  method 
of  derived  functions  is  considered  the  best  in  theory  (Professor  Shattuck). 

36.  (a)  Method  of  limits.    (&)  No. 

37.  (a)  Limits. 

38.  (a)  Method  of  limits.     (6)  No. 

39.  (a)  Limits.     (6)  No. 

40.  (a)  I  teach  the  infinitesimal,  prefer  it  in  general.  (&)  Occasionally  a 
student  will  not  accept  its  theories ;  I  then  try  him  on  limits  and  show  him 
their  relation. 

41.  (a)  The  latter,  with  a  sprinkling  of  the  former.  (J)  The  infinitesimal 
method  is  just  as  rigorous,  when  understood,  as  the  method  of  limits,  but  it  is 
my  experience  that  the  latter  more  quickly  removes  the  logical  difficulties  in 
the  way  of  the  beginner. 

42.  (a)  The  method  of  limits.  (6)  It  does,  provided  its  relation  to  the  method 
of  limits  be  shown ;  otherwise  not. 

43.  (a)  The  infinitesimal.  (Z»)  Students  have  generally  preferred  it  to  the 
method  of  limits. 

45.  (a)  The  infinitesimal.  (6)  It  does  when  it  is  known  that  results  do  not 
vary. 

46.  (a)  I  use  the  method  of  limits ;  the  method  of  infinitesimals  is  also  pre- 
sented,    (b)  One  method  seems  to  do  as  well  as  the  other  if  properly  presented. 

47.  (a)  Method  of  rates  (see  Taylor,  Eice  and  Johnson,  Buckingham). 

48.  (a)  Generally  by  limits.  (&)  To  the  first  part,  yes;  to  the  second  part, 
generally  not  very  satisfactory  to  those  going  over  the  subject  for  the  first  time. 

49.  (a)  The  infinitesimal.    (&)  Not  always. 


MATHEMATICAL   TEACHING   AT   THE   PRESENT   TIME.        313 

(o)  What  method  of  treating  the  calculus  do  you  favor,  that  of  limits,  the  infinitesimal,  or 
some  other?  (6)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student  ? — Continued. 

51.  (a)  Method  of  limits  ;  use  both.     (&)  Not  bo  rigorous  as  that  of  limits. 

52.  (a)  The  infinitesimal.     (5)  So  far  as  I  know  it  does. 

53.  We  have  no  classes  in  calculus. 

55.  (a)  The  calculus  is  not  a  part  of  our  course  of  study ;  personally,  I  prefer 
the  method  of  limits. 

56.  (a)  The  infinitesimal,     (b)  Yes. 

57.  (a)  This  is  elective — no  students  yet. 

58.  (a)  Have  used  the  integral. 

59.  (a)  Infinitesimal.  (&)  Yes,  when  the  student  is  well  drilled  in  what  should 
precede. 

60.  (a)  So  far,  the  method  of  limits.     (6)  Have  not  found  it  so,  generally. 

62.  We  do  not  teach  anything  higher  than  trigonometry. 

63.  (c)  That  of  limits.     (&)  No. 

64.  (a)  Infinitesimal,     (h)  Yes. 

65.  (a)  The  infinitesimal.  (6)  Not  entirely ;  but  the  ideas  of  the  calculus  are 
obtained  better  through  this  than  any  other  method. 

66.  (a)  That  of  limits,     (b)  Not  altogether  so. 

67.  (a)  The  two  methods  named  are  not  essentially  distinct ;  I  regard  the 
method  of  infinitesimals,  based  upon  the  doctrine  of  limits,  as  the  best  mode  of 
treating  the  subject.     (&)  Not  unless  it  is  based  upon  the  doctrine  of  limits. 

68.  (a)  The  method  of  rates,  passing  later  to  the  method  of  limits. 

69.  No  calculus. 

70.  (o)  Infinitesimal,     (h)  Not  entirely. 

71.  (a)  The  method  of  rates,  combined  with  the  method  of  limits.  (&)  It  is 
very  little  used,  and  only  after  the  others  have  been  taken  up. 

72.  (a)  Limits.  (&)  Students  have  seemed  satisfied  when  that  method  has 
been  used. 

73.  (a)  The  infinitesimal.     (6)  Healthful. 

74.  (o)  Infinitesimal  method  for  students  taking  brief  course.  (&)  Generally, 
yes. 

75.  (a)  The  infinitesimal.     (&)  By  sufficient  explanations. 

76.  (a)  Limiting  ratios  preliminary  to  the  more  direct  method  of  infinitesi- 
mals. (&)  Somewhat,  but  often  fundamental  investigations  are  made  more 
intelligible  to  the  beginner  by  this  method. 

78.  (a)  Limits.    (6)  It  does  when  properly  taught. 

79.  (a)  Limits.     (J)  No. 

80.  (a)  The  method  of  limits.  (J)  Yes,  if  the  student  persists  until  it  is  con- 
quered. 

81.  (a)  The  first  at  the  outset.  All  should  be  introduced  (see  Wundt, 
Logik). 

82.  (a)  That  of  limits.     (&)  Hardly. 

83.  (a)  Limits.     (&)  Yes. 

84.  (a)  Both  limits  and  infinitesimals.  (&)  Not  when  the  two  methods  are 
presented  together. 

85.  Calculus  is  not  studied  with  us. 

86.  (o)  Limits,  decidedly.  (J)  Not  until  the  student  has  mastered  the  method 
of  limits. 

■  87.  (a)  Infinitesimal.    (&)  Our  students  seem  to  understand  this  best. 

88.  (a)  That  of  limits. 

89.  (o)  Limits.    (6)  No. 

90.  (a)  The  infinitesimal  on  account  of  its  simplicity,  but  the  new  method 
by  General  C.  P.  Buckingham  is  excellent.    (6)  Not  always. 


314  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

(a)  What  method  of  treating  the  calculus  do  you  favor,  that  of  limits,  the  infinitesimal,  or 
some  other?  {i)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student?— Continued. 

91.  (a)  The  method  of  limits.    (&)  It  does  not. 

92.  (a)  Infinitesimal.     (6)  It  does. 

93.  (a)  The  flusionary  method.  (&)  Not  entirely;  it  is  taught  from  text- 
book— fluxionary  by  lectures. 

94.  (a)  Infinitesimal.     (&)  Yes. 

95.  (a)  Infinitesimal.     (&)  It  does. 

97.  (a)  The  German  method  of  limits,  not  the  popular  English  and  French. 

98.  (a)  The  rigorous  method  of  limits.  (&)  No;  there  is  an  evident  losa  of 
faith  at  this  point  for  students  on  first  reading. 

99.  (o)  We  use  both.     (&)  Yes. 

100.  (a)  Limits,     (i)  Do  not  teach  ifc. 

101.  (a)  Newtonian  fluxions.    (&)  No. 

102.  (a)  Limits.     (6)  No  ;  I  &nd  few  pupils  Satisfied  with  it. 

103.  (a)  Do  not  teach  calculus;  favor  limits. 

104.  (a)  We  use  both ;  sometimes  we  prefer  one  to  the  other. 

106.  (a)  The  infinitesimal  (ov 2)ractical  use,  but  that  of  rates  as  a  logical  basis. 
(b)  Not  as  satisfactory  as  the  theory  of  rates  as  given  by  Buckingham. 

107.  (a)  Limits  for  general  proof  and  infinitesimals  for  doing  examples.  (&) 
No,  not  alone. 

108.  (a)  Limits. 

109.  (a)  Limits. 

110.  (a)  The  method  of  limits  is  the  only  logical,  or  rational,  way  of  treating 
it;  though  the  infinitesimal  has  an  advantage  in  application.  (6)  No;  how 
a  quantity  can  have  another  quantity  taken  from  it  and  not  decrease  the  quan- 
tity so  diminished,  is  the  skeleton  that  will  not  down. 

112.  (a)  The  infinitesimal.     (6)  Yes;  both  rigorous  and  satisfactory. 

113.  (a)  The  method  of  limits.     (&)  Ifc  does  not. 

114.  (a)  Limits.     (6)  Not  perfectly. 

115.  (a)  The  method  of  infinitesimals,  if  i)roperly  taught.  (6)  Perfectly  so, 
■when  properly  taught  (Professor  Peck),  (a)  Limits.  (Z))  Only  when  explained 
in  connection  with  that  of  limits  (Tutor  Fiske). 

116.  (a)  Limits.    (&)  Not  unless  based  on  the  theory  of  limits. 

117.  (c)  Limits,     (b)  I  do  not  find  it  to. 

118.  (a)  We  use  infinitesimals,     (b)  Yes ;  in  general. 

119.  (a)  I  favor  the  method  of  fluxions,  but  use  the  infinitesimal,  mainly  be- 
cause I  could  not  get  a  suitable  text-book  in  fluxional  method  until  recently. 
(6)  Yes ;  better  than  the  method  of  limits ;  I  have  no  trouble  after  a  little  ex- 
planation. 

120.  (a)  The  method  of  rates  (the  Newtonian),  using  also  the  principles  of 
limits  in  connection  therewith.  (&)  No,  decidedly  no ;  if  not  established  by 
the  principles  of  limits. 

121.  (a)  The  infinitesimal  for  practical  use  ;  limits  as  a  means  to  an  end.  (6) 
Yes;  sufiSciently  so  for  all  practical  purposes. 

122.  (a)  Infinitesimals  for  use  ;  for  demonstration,  limits.     (&)  Yes. 

123.  (a)  Infinitesimal.    (&)  Yes. 

124.  (a)  We  teach  both  methods  simultaneously.  Having  understood  thor- 
oughly the  rigor  of  the  method  of  limits,  the  student  has  no  trouble  iu  hand- 
ling infinitesimals,  practically,  in  mechanical  problems.  (6)  Yes,  after  he  has 
once  thoroughly  understood  Taylor's  theorem,  not  as  a  formula  for  development 
in  series  merely,  but  as  the  means  of  determining  the  value  of  a  fanction  at  one 
point  from  its  value  at  another. 

125.  (a)  The  first,  but  the  infinitesimal  should  also  be  given  careful  state- 
ment. (&)  Yes,  if  properly  presented. 


MATHEMATICAL  TEACHINa  AT  THE  PRESENT  TIME.   315 

(a)  WJiat  method  of  ireating  the  calculus  do  you  favor,  that  of  limits,  the  infinitesimal,  or 
some  other  f  (h)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student  ? — Continued. 

126.  (a)  Infinitesimal.     (&)  Yea. 

127.  (a)  The  conception  of  calculus  as  the  "  science  of  rates.  (6)  To  some 
minds  it  is  not  satisfactory. 

128.  (a)  Limits,  supplemented  hy  the  conception  of  rates,  (b)  As  usually 
stated,  no. 

129.  {a)  The  infinitesimal.    (&)  Yes. 

130.  (a)  About  as  given  in  the  calculus  of  J.  M.  Taylor,  (h)  Not  in  the  ear- 
lier part  of  his  course,  hut  later.    Yes. 

131.  (a)  Limits.  (6)  The  students  find  it  easier,  and  most  collegi students  are  not 
very  critical. 

132.  (a)  We  use  Todhunter's  treatises,  who  employs  limits  in  the  diiferential 
calculus,  and  infinitesimals  in  the  integral  calculus,  and  we  find  it  to  work 
well.  (6)  Not  at  first,  hnt  later,  when  calculus  is  used  in  analytical  mechanics 
and  mathematical  physics,  it  carries  conviction  and  satisfaction. 

133.  (a)  The  infinitesimal.  (&)  It  may  seem  rigorous  at  first,  hut  I  think 
ultimately  he  is  better  satisfied  of  its  advantages  as  mental  drill. 

134.  (a)  Our  course  does  not  include  calculus. 

135.  (a)  Infinitesimal  in  the  main,  though  with  many  references  to  the  theory 
of  limits.    (6)  As  so  taught,  it  does. 

136.  (a)  I  prefer  the  infinitesimal  method,  but  I  do  not  hesitate  to  use  such 
assistance  as  can  be  derived  from  the  method  of  limits.  (&)  It  appears  to  me 
that  it  is  only  familiarity  with  the  proofs  that  makes  either  method  seem  rig- 
orous ;  but  the  difficulties  seem  no  greater  in  regard  to  one  than  the  other. 

137.  (a)  The  infinitesimal  for  beginners,  limits  for  the  advanced.  (J)  I  think 
it  does  if  properly  presented. 

138.  (a)  Infinitesimals  at  first,  afterward  the  method  of  limits  may  be  intro- 
duced.    (6)  With  99-100,  yes. 

139.  (a)  Limits  is  most  mathematical,  infinitesimals  most  easily  compre- 
hended.    (6)  It  is  satisfactory  to  ordinary  students. 

140.  (a)  Limits  for  the  theory,  infinitesimals  for  the  practice.    (6)  No. 

141.  (a)  Limits. 

142.  (a)  Infinitesimal. 

143.  (a)  I  hardly  know.  Each  has  its  advantages  and  disadvantages,  I  now 
use  the  method  of  rates  as  given  in  Eice  and  Johnson's  treatises.  (5)  It  docs 
not,  as  usually  presented. 

144.  (a)  Limits  for  theory,  inclusive  of  rates  and  of  infinitesimals.  (&)  It 
sometimes  seems  to  sa4fl.sify  the  students,  but  never  the  professor. 

145.  (a)  The  infinitesimal,  but  in  connection  with  the  other  methods.  (&)  It 
does,  if  any  method  does. 

146.  We  do  not  teach  calculus. 

147.  (a)  The  infinitesimal  at  first ;  I  use  both  to  some  extent.  (J)  Not  al- 
together. 

148.  (a)  That  of  flusions,  with  demonstrations  by  limits,  {h)  No ;  its  equa- 
tions (with  one  exception)  have  never  been  proved  true,  and  may  easily  be 
shown  false.  Its  results,  however,  are  absolutely  true,  as  experience  provea, 
and  as  may  be  shown  by  theory. 

149.  (a)  That  of  limits. 

150.  (a)  Infinitesimal.     (6)  It  does. 

151.  (a)  Limits.     (6)  It  does  not. 

152.  (a)  The  method  of  limits.     (&)  I  don't  use  the  method  in  teaching. 

153.  (a)  Infinitesimals  founded  upon  limits.  (&)  The  infinitesimal  method  ia 
never  rigorous  unless  founded  upon  limits. 


316  TEACHmG   AND   HISTORY   OF   MATHEMATICS. 

(a)  What  method  of  treating  the  calculus  do  you  favor,  that  of  limits,  the  infittitesi^nal,  or 
some  other?  (&)  Does  the  infinitesimal  seem  rigorous,  and  to  satisfy  the  mind  of  the 
student  f — Continued . 

154.  (a)  Limits-     (&)  It  is  the  practical  method,  but  is  not  satisfactory,  at 
first,  to  the  student. 

155.  (a)  I  am  decidedly  in  favor  of  the  method  of  limits.     (6)  It  does  not  seem 
rigorous,  and  does  not  satisfy  the  mind  of  the  student. 

156.  (a)  The  infinitesimal.     (&)  Perhaps  not  at  first. 

157.  (a)  That  of  limits. 

158.  (a)  Limits.     (6)  Yes;  can  he  presented  in  a  perfectly  satisfactory  man- 
ner. 

159.  (a)  The  theory  of  limits,  as  presented  by  Todhunter.     (J)  My  experi- 
ence with  classes  has  been  to  the  contrary.     It  does  not. 

160.  (a)  Limits.     (6)  It  has  not  done  so  with  my  classes. 

161.  (a)  By  limits.     (6)  It  does  not. 

163.  (a)  By  limits,     (b)  No. 

164.  (a)  The  method  of  rates  and  fluxions,  as  developed  by  Eice  and  John- 
eon.     (6)  No. 

165.  (a)  No  specialty. 

166.  (a)  Limits.     (6)  Yes. 

167.  (a)  The  method  of  rates,  combined  with  that  of  limits.    (&)  I  have  not 
been  able  to  make  it  as  real  to  my  students  as  I  desire. 

168.  (a)  The  infinitesimal  with  some  u  86  of  that  of  limits.     (&)Yes;  except 
with  a  few  students. 

(a)  Do  scientific  or  classical  students  shoio  greater  aptitude  for  mathematics  f   (b)  TVliich 

sex  ? 

1.  (a)  No  special  difference  is  noticed. 

2.  Classical; 

3.  Scientific. 

5.  Classical. 

6.  (&)  Male. 

8.  (a)  No  difference  ;  our  Civil  Engineering  students  have  shown  most  apti- 
tude.    (6)  Male  sex. 

9.  (a)  I  tbink  it  depends  on  talent. 

10.  (6)  Male,  generally, 

11.  (h)  Girls  for  rote  work,  boys  for  original  work. 

12.  (a)  Classical.  ■  (t)  Male. 

15.  (a)  I  observe  little  or  no  difference. 

16.  (a)  Classical.     (6)  Male. 

17.  (t)  Male. 

20.  (a)  Scientific,     (b)  Male. 

21.  (a)  Classical,     (b)  With  us,  females. 

23.  (a)  Scientific. 

24.  (a)  Scientific.     (6)  Equal. 

25.  (a)  Scientific.     (6)  Male. 

26.  (a)  Scientific.     (&)  Boye. 

27.  (a)  Scientific,  as  a  rule. 

28.  (a)  Scientific,     (b)  Male. 

29.  (6)  I  note  no  material  difference  in  our  work. 

30.  (a)  I  am  inclined  to  think  scientific.     (6)  Male,  but  sometimes  female. 

31.  (a)  Have  not  been  able  to  note  great  difference.    (&)  In  my  classes  tho 
young  ladies  have,  as  a  rule,  excelled. 

32.  (a)  Scientific.    (&)  The  male. 


MATHEMATICAL   TEACHING   AT   THE   PRESENT   TIME.         317 

(a)  Do  scientifiG  or  classical  students  sJioto  greater  aptitude  for  mathematics  f    (&)  Which 

sex  ? — Continued. 
33.  (a)  The  greater  the  drill,  the  greater  the  aptitade  for  anything.    There- 
fore elassical.     (6)  The  young  men  for  persistent,  the  young  ladies  for  instau- 
taneous  grasp. 

35.  (a)  Scientific.     (6)  Male. 

36.  (a)  Good  students  in  either  course.     (6)  Males  as  a  rule. 

37.  (a)  Classical,     (b)  Males. 

38.  (a)  I  would  say  classical.     (&)  There  are  more  males  than  females. 
40.  (a)  Difficult  to  answer.    (6)  The  ladies  are  not  inferior. 

42.  (a)  Generally  the  scientific,     (h)  1  see  no  difference. 

43.  (a)  No  difference  noted.  (6)  Males  average  higher,  but  a  female  often 
stands  first. 

44.  (6)  A  greater  number  of  males  succeed,  but  a  few  females  excel. 

45.  (a)  Cannot  see  any  difference. 

46.  (a)  Classical  students  not  required  to  study  analytical  mathematics.  No 
comparison  is  possible,  aa  we  have  no  preparatory  school.  (&)  Young  men 
form  the  larger  part  of  our  higher  classes.  As  far  as  comparison  is  possible,  the 
two  sexes  show  about  equal  aptitude  foe  the  study. 

47  (a)  Our  best  have  been  classical  here,  but  thereversehasbeenmy  experi- 
ence elsewhere.     (&)  Girls  in  the  test-book ;  boys  outside. 
•  49.  (a)  No  diflerence.    (6)  Male. 

51.  (a)  There  is  no  difference,  (b)  Some  of  our  best  students  in  mathematics 
have  been  young  ladies. 

52.  (a)  No  difference.     (6)  Interest  about  equal. 

53.  (a)  The  scientific.     (6)  Male. 

54.  (a)  Generally  scientific,     (b)  About  equally. 

55.  (a)  The  former.     (&)  No  perceptible  difference  as  to  sex. 

56.  (a)  Classical,     (b)  Gentlemen. 

57.  (a)  Generally  the  former. 

59.  (b)  Equal  in  applied,  but  more  males  in  pure. 

60.  (a)  Scientific. 

61.  (a)  Classical. 

63.  (b)  Male. 

64.  (a)  Scientific. 

65.  (a)  Scientific,     (b)  Male 

66.  (a)  Classical  students. 

73.  (a)  Scientific      (b)  No  difference. 

75.  (a)  Good  students  in  other  departments  are  equal  in  mathematics  as  a 
rule,  (b)  It  is  rather  difficult  to  answer  directly.  The  ladies  average  fully  as 
high  as  the  gentlemen. 

76.  (a)  Scientific,     (b)  Male. 

77.  (a)  Classical,     (b)  Male. 

79.  (fl.)  Classical,  thus  far. 

80.  (a)  Classical. 

82.  (a)  Scientific,  usually,    (b)  I  find  little  difference. 

83.  (a)  Classical. 

84.  (b)  The  male  sex. 

85.  (a)  The  classical,  as  a  rule ;  our  very  best  mathematical  students  have 
been  scientific. 

86.  (a)  No  difference. 

87.  (a)  No  classical,     (b)  Young  men. 

88.  (a)  Classical. 

90.  (a)  Scientific,     (ft)  Males. 

91.  (a)  Scientific,     {b)  Male  for  analysis,  female  for  book  work. 


318  TEACHING  AND   HISTOEY   OF   MATHEMATICS. 

(o)  Do  scientifio  or  classical  students  show  greater  aptitude  for  mathematics?    (6)  Which 

sex  ? — Continued. 

92.  (a)  Scientific  or  philosopbical.  (6)  Nearly  alike  on  the  text-book  work. 
The  gentlemen  seem  more  successful  in  original  investigation.  May  not  the 
reason  for  this  be  found  in  the  fact  that  it  has  been  assumed  for  an  indefinite 
period  that  woman  is  not  capable  of  doing  such  work,  and  so  she  has  not  been 
required  to  do  it,  thus  leading  to  a  dwarfing  of  this  part  of  her  mind  ? 

93.  (a)  Seem  not  to  divide  on  this  line.  (&)  See  no  difference.  Few  girls 
elect  calculus. 

94.  (a)  No  appreciable  difference.  (&)  The  females  do  closer  work  ou  lessons 
and  tasks  assigned. 

95.  (a)  Scientific. 

97.  (a)  Classical,  and  students  of  the  exact  (not  merely  scientific)  eciences. 
(I)  Male. 

98.  (a)  Scientific.     (&)  Male.      , 

99.  (a)  Scientific. 

100.  (5)  Male. 

101.  (a)  Scientific.     (6)  Male. 

102  (a)  The  scientific.    (6)  Males  in  quantity,  females  in  quality. 

104.  (a)  No  difference,  high  classics  generally  carry  high  mathematics. 

105.  (a)  No  difference,     (b)  Hard  to  say.    Boys  a  little  better  reasonera. 

106.  (a)  Classical.     (&)  On  the  whole,  young  men.  • 
108.  (a)  Can't  say.     (6)  Males. 

113.  (a).  Scientific. 

114.  (a)  Classical  students  frequently  show  a  greater  aptitude,  but  scientific 
students,  after  having  a  practical  end  in  view,  more  frequently  become  accom- 
plished mathematicians. 

115.  (a)  Scientific  (Professor  Peck),     (a)  Mixed.     (&)  Male  (Tutor  Fiske). 

116.  (a)  Classical  generally. 

117.  (a)  Scientific,     (b)  Male. 

118.  (a)  We  see  little  difference. 

119.  (a)  Classical,     (b)  About  equal. 

121.  (a)  Scientific. 

122.  (a)  Scientific. 

123.  (a)  Eather  the  classical.     (&)  Young  men. 

124.  (a)  Scientific.  The  mathematics  in  the  classical  course  ends  with  trig- 
onometry, (b)  Only  a  few  girls  take  mathematics.  Can't  answer  satisfac- 
torily. Some  girls  do  excellent  work.  I  doubt  whether  sex  has  much  to  do 
with  natural  mathematical  ability. 

125.  (a)  Other  conditions  being  equal,  no  difference.  (6)  No  difference,  fol- 
lowing similar  preliminary  training. 

127.  (a)  Difficult  to  decide,     (b)  Not  much  difference. 

128.  (a)  Our  experience  not  a  fair  test — students  have  been  so  largely  clas- 
sical.    (6)  Male,  on  the  whole. 

129.  (a)  Classical. 

130.  (a)  Scientific. 

131.  (a)  About  the  same.     (&)  Male. 

132.  (a)  No  difference  in  aptitude,  but  classical  students  find  no  time  for  ex- 
tended courses.  (6)  Comparatively  few  females  excel,  though  some  are  as 
good  as  any  of  the  males. 

133.  (o)  Very  little  difference.     (Z))  Males. 

135.  (a)  Scientific  students,     (b)  Male. 

136.  (a)  Classical  students.     (&)  No  experience. 

137.  (a)  Scientific  or  technical. 

138.  (a)  So  far  as  we  can  test  it,  classical. 


I 


MATHEMATICAL   TEACHING   AT    THE   PRESENT   TIME.        319 

(a)  Do  scientific  or  classical  students  shoto  greater  aptitude  for  inathetnatics  f    (&)  Which 

sex  ? — Continued. 

139.  (a)  No  difference  in  aptitude.     (6)  No  difference  in  sex, 

140.  (a)  Classical  are  more  frequently  the  best. 

142.  (a)  Classical.     (&)  The  male  by  a  large  percentage. 

144.  (a)  Classical. 

145.  (o)  Scientific. 

146.  (a)  I  notice  no  difference. 

147.  (a)  The  classical  are  superior,     (b)  No  difference. 

148.  (a)  The  scientific,  but,  for  the  reason  (I  think)  that  those  who  dislike  it 
elect  a  literary  course. 

149.  (a)  Classical,  I  think.  (6)  Male  and  female  eq^ually  is  my  obeervatioa 
here. 

150.  (a)  Classical.     (6)  Males. 

151.  (a)  Classical.     (6)  Male  sex. 

153.  (a)  Scientific.     (&)  Male. 

154.  (a)  We  have  no  classical  students. . 

155.  (o)  Scientific. 

156.  (a)  Scientific. 

157.  (a)  Scientific. 

159.  (a)  Scientific. 

160.  (a)  Scientific. 

161.  (a)  The  classical.     (6)  Male. 

162.  (a)  I  think  classical,     (b)  Male. 
164.  (a)  Scientific.     (6)  Male. 

166.  (a)  Classical.     (&)  I  do  not  see  any  difference. 

167.  (a)  Generally  the  scientific.  (6)  Males  as  a  rule,  at  least  in  higher 
branches  than  elementary  trigonometry. 

168.  (a)  The  classical. 

(o)  In  what  other  subjects  are  good  mathematical  students  most  successful?    (6)  In  what 

least  successful? 

I.  (a)  Good  students  in  mathematics  generally  stand  well  in  all  other  studies. 

4,  (a)  As  a  rule  none  but  the  better  students  pursue  mathematics  more  than 
two  years.  Good  mathematical  students  are  successful  in  the  scientific  branches 
taught  here.    (6)  In  English. 

5.  (a)  Geography,  logic,  history,  chemistry,  and  natural  philosophy.  (6) 
Grammar,  rhetoric. 

8.  (a)  In  logic  and  in  the  physical  sciences.  (&)  Ancient  languages  and  Eng- 
lish literature. 

9.  (a)  In  philosophy,  chemistry,  analytical  mechanics,  geodesy,  etc.  (6)  I 
have  npt  observed. 

10.  (a)  I  can  not  say.     (&)  Do  not  know  ;  hard  to  determine. 

II.  (a)  Am  not  certain ;  should  say  physics  and  ancient  languages.  (6)  Can't 
say. 

12.  (c)  I  do  not  know  certainly,  but  I  have  never  noticed  any  inverse  relation 
between  linguistic  and  mathematical  endowments.  Chemistry  and  mathemat- 
ics are  less  friendly. 

15.  (a)  Scientific  subjects,  especially  physics.  But  I  observe  that  good  mathe- 
maticians usually  do  well  in  almost  any  subject  which  interests  them.  (&)  Sub- 
jects which  involve  much  "committing  to  memory." 

16.  (a)  A  good  student  is  successful  everywhere.  I  have  found  that  my  best 
students  in  mathematics  were,  as  a  rule,  "  best  students  "  in  other  departments. 

17.  (a)  In  any  subject  in  which  continued  reasoning  is  necessary,  (b)  I  am 
unable  to  specify. 


320  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

(a)  In  tohat  othei'  suhjects  are  good  mathematical  students  most  successful  f    (i)  In  what 
least  successful  f — Continued. 

18.  (a)  In  natural  philosophy,  in  metaphysics,  and  generally  in  Greek.  (6) 
Literature. 

19.  (a)  They  do  equally  well  in  Latin,  as  a  rule.     (&)  English. 

20.  (a)  Mechanics. 

21.  (a)  A  student  good  in  mathematics  is  apt  to  he  successful  in  all  hranches. 

22.  (a)  My  experience  goes  to  show  that  a  student  who  is  good  in  mathemat- 
ics is  capable  of  coming  off  with  good  standard  in  almost  any  other  study.  I 
have  known  a  few  apparent  exceptions,     (b)  In  languages. 

24.  (a)  Sciences.    (&)  Classical  studies. 

25.  (o)  Physics. 

26.  (o)  Physics,  logic,  chemistry,     (h)  Ehetoric. 

27.  (a)  In  engineering,  physics,  astronomy.  Mathematical  training  seems  to 
make  lawyers  more  successful  in  the  clear  statement  of  their  cases.  (6)  Liter- 
ary pursuits. 

28.  (a)  In  mental,  moral,  and  natural  philosophy,     (b)  In  belles-lettres. 
30.  (a)  In  almost  every  other,     (ft)  Perhaps,  literature. 

32.  We  can  hardly  give  an  intelligent  answer  to  this  question  with  our  grade 
of  work. 

33.  (a)  Mathematics,  as  we  are  compelled  to  teach  it,  is  largely  mechanical; 
therefore,  in  subjects  not  requiring  great  originality.     (6)  Answered. 

35.  (a)  In  engineering.     (6)  In  languages. 

37.  (o)  Varies  with  the  student. 

38.  As  a  rule,  our  mathematical  students  are  excellent  in  all  their  studies. 
Languages  are  not  unfrequently  hard  for  good  mathematical  students. 

39.  (a)  Greek,  Latin — often — mainly — various  forms  of  graphics.  (6)  Scien- 
tific research,  i.  e.,  natural  sciences. 

40.  (a)  The  various  branches  of  natural  science,  metaphysical  studies. 

41.  (a)  Usually  in  all  other  subjects  of  our  course.  (6)  Occasionally  in  lan- 
guages. 

43.  (a)  We  often  have  iine  work  upon  topics  related  to  general  geometry. 

44.  (a)  Chemistry,  physics,  languages,     (b)  History,  literature. 

45.  (a)  Generally  in  whatever  is  undertaken,  I  believe  success  in  any  branch 
is  in  proportion  to  application. 

46.  (o)  Whatever  they  undertake.  (6)  Whatever  they  give  the  least  atten- 
tion to. 

47.  (a)  Logical,     (h)  Linguistic. 

48.  (a)  As  a  rule  in  all  subjects  requiring  judgment,  reason,  discrimination. 
(b)  In  subjects  requiring  the  memory  as  the  chief  element  of  the  mind. 

49.  (a)  Sciences,     (b)  Languages. 

50.  (a)  Languages.     (&)  History. 

51.  (a)  All  others,  that  is,  according  to  circumstances. 

52.  (a)  Our  good  mathematical  students  are  good  in  languages  and  sciences. 

53.  (a)  Chemistry  and  physics.     (6)  Have  not  noticed. 

54.  (a)  Some  in  one  subject  and  some  in  another,  according  to  native  aptitude 
and  application. 

55.  (a)  In  chemistry,  jihysics,  and  logic.  Good  mathematical  students  rarely 
show  weakness  iu  any  study.     (&)  Literature  (and  modern  languages?). 

56.  (a)  Natural  science,     (b)  Language. 

57.  (a)  Natural  science. 

59.  (a)  The  majority  of  good  mathematical  students  are  good  in  everything 
else,  but  sometimes  a  mathematical  mind  fails  in  letters,  and  vice  versa,  as  some 
appreciate  only  demonstrative  reasoning,  and  some  moral. 

60.  (o)  Physics,  astronomy^  and  natural  science.     (6)  Languages. 


MATHEMATICAL  TEACHING  AT  THE  PEESENT  TIME.    321 

(a)  In  tvhat  other  subjects  are  good  mathematical  students  most  successful  ?    (J)  In  what 
least  successful  ? — Continued. 
63.  (a)  I  find  good  mathematical  students  suocessfulj  generally,  in  all  other 
subjects. 
66.  (a)  In  most  subjects. 

68.  (a)  Our  best  students  are  about  equally  successful  in  all  mathematical 
branches. 

69.  (a)  Natural  science.     (&)  Can't  say. 

70.  (a)  Languages.     (&)  History  and  literature. 

73.  (a)  Physics  and  astronomy.  (&)  My  best  mathematicians  are  best  in 
other  lines, 

74.  (a)  Since  I  have  observed  here  (four  years),  the  best  mathematical 
students  are  usually  also  among  the  best  in  all  studies ;  otherwise  in  natural 
sciences,  English  and  Greek,  history  and  political  science,     (b)  Languages. 

7.5.  (a)  The  good  mathematicians  are  those  whose  general  standing  is  high, 
but  of  course  there  are  esceptions  to  this ;  I  should  say  that  they  are  more  likely 
to  excel  in  the  sciences,  logic,  and  metaphysics. 

76.  (a)  Chemistry,  logic,  mental  science,  (i)  Language,  history,  rhetoric, 
oratory. 

79.  (a)  Languages,  I  should  say,  in  general. 

80,  (a)  Latin  and  science.     (&)  History  and  literature. 

82.  (a)  As  a  rule,  I  think,  in  all  subjects,  although  occasionaDy  I  find  one 
■who  is  weak  in  language  and  literary  studies. 
84.  (a)  Physics  and  mechanics.     (&)  The  languages. 

86.  (a)  No  special  diiference  so  far  as  I  know. 

87.  (a)  Chemistry,  physics,  and  applied  mechanics.     (&)  Languages. 

88.  (a)  Our  records  show  that  good  mathematical  students  are  successful  in 
all  other  subjects, 

89.  (a)  In  all  other  subjects  taught  in  the  school. 

90.  (a)  In  historical  studies,  natural  philosophy,  and  mathematical  astron- 
omy.    (6)  Literary,  but  not  always, 

91.  (a)  In  the  lateral  sciences,  e.  g.,  physics,  chemistry;  also  in  logic.  (6) 
Languages  and  history. 

92.  (a)  As  a  rule,  I  think,  in  the  sciences,  and  especially  in  original  investiga- 
tions in  science.     (&)  So  far  as  my  observation  goes,  in  languages,  as  a  rule. 

93.  (a)  With  rare  exceptions  they  are  good  in  all  the  subjects.  The,  converse 
is  not  so  general,  I.  e.,  students  often  excel  in  one  or  two  departments  without 
excelling  in  mathematics. 

94.  (a)  They  average  well  all  around.     (&)  No  uniformity, 
^5.  (a)  Draughting,  physics,  chemistry,  logic. 

96.  (a)  As  a  rule,  those  good  in  mathematics  are  good  in  all  others,  but  es- 
pecially in  natural  sciences,  psychology,  and  logic. 

97.  (a)  In  all  the  more  introspective,  and  such  as  require  prolonged  and  stren- 
uous thought,  not  mainly  observation  (like  stone — or  bug — lore),  (b)  In  these 
latter  so-called  experimental  sciences. 

98.  (a)  Applied  arts,  engineering,  physics,  etc,  (6)  Languages,  metallurgy, 
analytical  chemistry, 

99.  (a)  Philosophy.     (&)  Composition. 

101.  (a)  Philosophical.     (&)  Linguistic, 

102.  (a)  In  natural  sciences,  history,  geography,  logic.     (6)  Languages. 
104.  (a)  All  scientific  pursuits,  drawing,  arts,  generally. 

106.  (a)  In  such  as  require  concentration  of  mind,  and  close  reasoning,  (b) 
If  in  any,  in  such  as  depend  upon  observation  and  experiment. 

107.  (a)  Generally  in  all  others,  if  they  are  interested. 

109,  (a)  Natural  philosophy,  chemistry,  Greek,  Latin.    They  can  generally 
do  well,  wherever  they  try.    (6)  English,  political  science. 
8,S1_^^o.  8 21 


322 


TEACHING  AND   HISTORY   OF   MATHEMATICS. 


(a)  ImvTiat  oilier  subjects  are  good  matJiematical  students  most  successful  f    (6)  InwT'nt 
least  successful  ? — Continued. 

110.  (a)  My  observation  has  been  that  where  students  were  good  in  matlie- 
matics,  they  were  good  in  all  their  other  studies. 

111.  (a)  Logic  and  analytical  studies. 

112.  (a)  They  generally  stand  high  in  all  subjects. 

113.  (a)  As  a  rule  they  are  successful  in  all  other  studies;  more  so  in  meta- 
physics, theology,     (b)  The  higher  study  of  literature. 

114.  (a)  Very  difficult  to  generalize.  Many  excellent  mathematicians  are 
"  all-around "  men.  Others  excel  in  science,  and  are  least  successful  in  lan- 
guages and  speculative  subjects. 

115.  (a)  A  good  student  in  mathematics  is  generally  a  good  student  in  all 
other  branches  (Professor  Peck). 

116.  (a)  Generally  in  logic  and  psychology. 

117.  (a)  Mechanics,  physics,  chemistry,  logic.     (6)  Classics. 

118.  (a)  They  are  generally  good  in  all  subjects.  (6)  Subjects  requiring 
memory  only. 

120.  (a)  Chemistry  (including  heat,  physiology,  electricity,  and  magnetism), 
mineralogy  and  geology,  engineering,  ordnance  and  gunnery,  andlaw.  (&)  Draw- 
ing, Spanish,  and  French,  relatively.  Good  mathematical  students  are  gener- 
ally good  in  all  other  branches  (Professor  Bass,  professor  of  mathematics). 

Charles  W.  Larned,  the  professor  of  drawing  at  West  Point,  answers  as  fol- 
lows :  I  differ  somewhat  from  the  inferences  to  be  drawn  from  the  answer  to 
this  question  by  the  professor  of  mathematics. 

In  so  far  as  any  influence  is  to  be  implied  by  mathematical  proficiency  upon 
other  studies,  an  examination  of  the  standing  of  the  last  five  graduating  classes 
tends  to  show  very  positively  that  law  belongs  to  the  category  of  those  studies 
in  which  there  exists  the  greatest  discrepancy,  and  this,  notwithstanding  that 
law  is  studied  two  years  after  mathematics  is  completed  and  when  habits  of 
study  and  ability  to  master  a  wider  range  of  subjects  is  more  highly  developed 
by  the  study  of  intermediate  synthetic  studies. 

There  is  a  much  greater  range  of  discrepancies  also  between  the  group  of 
studies  comprised  under  the  head  of  chemistry  (which  includes  electrics,  min- 
eralogy and  geology,  and  heat)  and  mathematics  than  would  naturally  be  in- 
ferred from  the  grouping  made.  Even  in  natural  philosophy  the  aggregate 
discrepancies  were  greater  than  I  had  supposed  probable. 

The  standings  of  the  graduating  classes  of  1888,  1887,  1886,  1885,  and  1884  in 
law,  chemistry,  drawing,  English,  and  French  were  reduced  to  the  same  stand- 
ard and  the  differences  between  these  and  mathematics  in  each  case  were  ob- 
tained, and  the  aggregate  in  each  subject  for  these  classes  is  as  follows : 

Discrepancies  as  compared  tvith  mathematics. 


1888 
1887 
1886 
1885 
1884 


Law. 


356 
925 
1,102 
336 
195 


Chemistry. 

Drawing. 

French. 

342 

481 

418 

714 

1,021 

936 

1,096 

],694 

1,166 

250 

332 

295 

144 

366 

254 

English. 


413 
996 
1,376 
301 
241 


In  regard  to  drawing  it  is  proper  to  observe  that  a  marked  distinction  exists 
between  technical  graphics  andfree-hand  drawing.  The  standing  given  includes 
ftee-hand  drawing,  occupying  one-fourth  of  the  courae.    In  this  the  possession 


MATHEMATICAL  TEACHING  AT  THE  PEESENT  TIME.    323 

(a)  In  what  other  subjects  are  good  mathematical  students  most  successful  f  (&)  In  what 
least  successful? — Continued. 
of  natural  graphical  talent  exercises  a  much  greater  influence  in  producing 
discrepancies  in  standing.  In  technical  graphics,  however,  throwing  out  a 
few  men,  perhaps  ond-half  dozen  in  each  class,  with  pronounced  natural  ability, 
standing  in  plane  and  descriptive  geometry  has  a  decidedly  beneficial  influence 
on  standing  in  drawing.  In  other  respects,  intelligence,  whether  mathematical 
or  liberal,  will  tell  in  the  work.  Leaving  out  four  or  five  exceptional  men  in 
each  class  the  discrepancies  in  drawing,  even  with  free-hand  included,  fall 
below  those  of  French,  English,  and  law. 
12L  (a)  Physics  and  astronomy.     (&)  Latin,  Freiich,  etc. 

122.  (a)  First-rate  mathematical  students  generally  do  well  in  all  other 
studies. 

123.  (a)  In  mathematics  of  physics.  (6)  Possibly  the  biological  sciences  and 
languages. 

124.  (a)  My  experience  is  that  a  man  who  is  good  in  mathematics  has  mental 
ability  sufficient  to  make  any  subject  of  an  ordinary  college  course  compara- 
tively easy.   Good  in  mathematics — good  everywhere. 

125.  {a)  Applied  sciences,  of  cours,e — astronomy,  physics,  logic,  and  meta- 
physics, (h)  Languages  and  literature,  sometimes.  Still  hardly  think  that  is 
true,  as  a  rule. 

126.  (a)  Advanced  work  in  physics  and  engineering.  (6)  Our  best  students 
in  mathematics  are  best  everywhere. 

129.  (a)  Good  mathematical  students  are  good  in  all  their  work.  (&)  Earely 
unsuccessful  in  any  line  of  study. 

130.  (a)  Physics,  astronomy,  logic. 

131.  (a)  Classics,  sciences,  but  there  are  many  exceptions,  (b)  English, 
probably. 

133.  {a)  Think  that,  on  the  whole,  our  best  mathematical  students  are  best, 
generally,  in  other  studies. 

134.  (a)  Sciences. 

136.  (a)  Greek  is  often  combined  with  mathematics. 

137.  (a)  They  are  generally  good  all  around.  (6)  In  languages,  but  only  in 
exceptional  cases. 

138.  (a)  In  all  other  subjects.     (&)  None. 

140.  (a)  Physics  and  astronomy.  The  good  ones  are  also  usually  good  in 
classics  and  everything. 

141.  (a)  Generally  also  in  the  classical  studies.' 

142.  (a)  Physics,  chemistry,  logic.     (6)  Moral  and  mental  philosophy. 

143.  (a)  All  subjects  requiring  accurate  thought.  In  our  college  this  is  es- 
pecially noticeable  in  mental  and  moral  philosophy,  (b)  Those  requiring 
mere  memory. 

144.  (a)  Usnally  in  all  others, 

145.  (a)  Some  in  one,  some  in  another.  No  general  rule,  (b)  No  general 
rule. 

146.  (a)  In  this  they  differ,  though  they  are  possibly  better  in  scientific 
studies.     (&)  Languages. 

148.  (a)  I  have  not  observed  that  a  successful  student  in  mathematics  is 
more  apt  to  succeed  in  one  subject  than  he  is  in  another,  except  where  the  sub- 
ject rests  on  mathematics. 

150.  (a)  Generally  in  everything  else. 

151.  (&)  Belles-lettres. 

152.  (a)  Naturally  in  subjects  depending  upon  a  knowledge  of  mathematics, 
and  generally  in  whatever  else  they  may  study. 

153.  (a)  In  logic,  physics,  engineering,  medicine.     (&)  Languages. 


324  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

(a)  In  what  other  subjects  are  good  mathematical  students  most  successful?    In  what  least 

successful  ? — Continued. 

154.  (a)  In  all  branches  that  require  accurate  observation  and  close  reason- 
ing.    (6)  In  languages. 

155.  (a)  In  branches  of  natural  science.     (5)  In  the  languages,  I  think. 

156.  (a)  In  logic  and,  as  a  rule,  the  natural  sciences. 

157.  (a)  They  are  apt  to  be  more  successful  in  all  subjects  connected  with  the 
sciences  than  in  the  study  of  languages. 

158.  (a)  As  a  rule,  all  students  standing  very  well  in  mathematics  will  achieve 
success  in  any  other  subject.    I  have  seen  but  few  exceptions. 

159.  (a)  In  moral  and  mental  philosophy,  logic  and  civil  law.  (J)  Synthetic 
languages. 

160.  (a)  In  the  mathematical  sciences.  A  good  mathematical  student  is  good 
at  everything  he  undertakes. 

161.  (a)  Physics  and  chemistry.     (&)  Languages  and  history. 

162.  (a)  Oftener  classics. 

163.  (a)  Engineering  and  physics,     (h)  Languages. 

164.  (a)  In  any  subjects  requiring  reflection.  (Z>)  Those  requiring  perception 
and  memory  only. 

166.  (a)  Logic,  chemistry,  philosophy,  political  economy,  and  astronomy. 
(J))  Language. 

167.  (a)  I  believe  that  a  really  first-class  mathematical  student  is  generally 
successful  in  nearly  all  subjects,  but  those  a  grade  lower  are  most  likely  to 
excel  in  the  physical  sciences  than  in  other  lines.     (6)  Perhaps  in  belles-lettres. 

168.  (a)  I  don't  think  I  can  tell,  for  there  is  such  diversity ;  yet  I  think  that 
those  who  are  good  in  mathematics  are  good  all-around  students,  as  a  rule. 

What  is  the  relative  prominence  of  maiheviatics  in  your  course  of  study  as  shoton  hy  hours 

pel'  xoeeli  and  per  year  ? 

1.  In  classical  and  scientific  courses  the  same  number  of  hours  is  given  to 
mathematics  as  to  any  other  study.  In  the  engineering  course  about  twenty- 
five  per  cent.  more. 

2.  About  one-fourth  part  of  the  class-time  is  devoted  to  the  study  of  mathe- 
matics. 

3.  No  study  has  more  attention,  and  some  have  less. 

4.  Five  hours  per  week  out  of  sixteen  hours  for  recitation  and  lectures  are 
devoted  to  mathematics  for  three  years ;  the  last  year  three  hours  per  week 
during  the  year. 

5.  More  prominence  given  to  mathematics  than  to  any  other  study. 

6.  Large. 

7.  Occupies  more  time  than  any  other  subject. 

8.  In  classical  course  one-fourth  of  student's  time  is  devoted  to  mathe- 
matics ;  relatively  more  in  scientific  and  civil  engineering  course. 

9.  The  principle  studies  in  our  college  receive  equal  attention  ;  mathematics 
one  hour  aud  a  quarter  daily. 

10.  More  stress  on  mathematics  as  a  whole  than  upon  any  other  subject,  I 
think. 

11.  Mathematics  takes  one-fourth  of  the  time  in  the  scientific  course,  one- 
seventh  in  the  literary,  and  over  one-fifth  in  the  classical. 

12.  It  leads  Greek,  and  is  on  a  par  with  Latin  and  physics. 

13.  Considered  of  fundamental  importance  and  continued  throughout  the 
four  yeavs  of  study ;  twenty  hours  per  week  for  the  four  classes  (pure  mathe- 
matics only). 

14.  During  the  year,  four  hours  per  week  out  of  a  total  of  fifteen  hours. 

15.  In  the  classical  course  about  ten  per  cent,  of  the  work  is  in  mathematics, 
and  in  the  scientific  course  about  fifteen  per  cent.  I  have  counted  only  the 
prescribed  work  and  the  pure  mathematics,  so-called. 


MATHEMATICAL  TEACHING  AT  THE  PEESFNT  TIME.    325 

What  is  the  relative  prominence  of  mathematics  in  your  course  of  study  as  shown  iy  hours 
per  iveeTc  and  per  year? — Continued. 

16.  It  has  the  same  prominence  as  do  the  classics. 

17.  Preparatory,  two-thirds  of  the  entire  worV;  done  is  mathematics ;  first  year, 
one-half;  second,  about  two-fifths ;  third,  one-fifth;  fourth,  only  applied  math- 
ematics. 

18.  One  hour  of  mathematics  each  day,  i.  e.,  six  hours  per  week ;  about  three 
and  three-fourths  of  other  studies. 

19.  Our  regular  course  of  study  practically  covers  five  years,  divided  into 
three  terms  each;  mathematics  occupying  one-third  of  each  term  for  the  ten 
terms  ending  with  first  term  of  the  Junior  year  (Professor  Gordon).  In  the 
early  years  of  the  course,  equal  to  any  other  subject  except  English  (Professor 
Draper). 

20.  Mathematics  and  classics  each  occupy  five  times  as  many  hours  as  sci- 
ence. 

21.  Twenty  per  cent. 

22.  Daily  recitation  required  of  every  student. 

23.  It  is  of  the  first  prominence. 

24.  No  special  prominence  observed. 

25.  Six  hours  i)er  week ;  about  one  hundred  and  eighty  per  year. 

26.  One-fifth  of  time. 

27.  Mathematics  has  seventeen  hours  per  week ;  the  ancient  languages,  fifteen 
hours  per  week ;  English,  twelve  hours  per  week. 

28.  One-fourth  of  all  the  time  during  Freshman  and  Sophomore  years  is  de- 
voted to  pure  mathematics ;  and  one-tenth  of  all  the  time  in  the  Junior  and 
Senior  years. 

29.  Mathematics  extends  through  'two-thirds  of  the  course.  Takes  about  one- 
fourth  of  time  during  that  period. 

30.  The  time  spent  is  about  the  same  as  in  the  average  college. 

31.  More  time  is  given  than  to  any  other  one  topic. 

32.  It  ranks  with  the  natural  sciences  and  the  ancient  and  the  foreign  lan- 
guages. 

33.  As  prominent  as  any  other  chair,  if  not  more  so. 

34.  Freshman  year,  three- fifteen ths ;  Sophomore  year,  three-fif  teenths  ;  Junior 
year,  two-fifteenths  (elective) ;  Senior  year,  two-fifteenths  (elective). 

35.  In  some  courses  one-third  the  time  for  one  year ;  in  others  one-third  the 
time  for  seven  terms  out  of  twelve,  with  applied  mathematics  for  eleven  terms 
more.    In  both,  eighteen  units  out  of  thirty-six. 

36.  Nearly  one-third  in  Freshman  and  Sophomore  years. 

37.  The  same  as  other  studies;  three  and  three-fourths  hours  per  week. 

38.  One  year's  work  is  required  of  all  students.  Four  years  are  required  of 
mathematical  students.    Many  elect  mathematics  for  one  or  more  years. 

39.  The  time  is  about  equally  distributed  between  Latin,  Greek,  modern 
languages,  and  mathematics. 

40.  Different  in  the  various  courses.  Mathematics  is  more  prominent  in  the 
scientific  course,  claiming  about  one-fourth  the  student's  time  (perhaps  one- 
third). 

41.  Freshman  year  twenty-one  hours  out  of  sixty-one,  798  per  year;  Sopho- 
more eighteen  out  of  fifty-four,  684  hours  per  year  ;  Junior  thirteen  out  of 
sixty,  494  per  year.  In  this  estimate  two  hours  of  preparation  are  usually 
reckoned  with  each  hour  of  recitation. 

42.  Sixteen  hours  per  week. 

43.  Upon  an  equality  with  Latin  and  Greek. 

45.  It  is  taught  five  hours  per  week  until  the  end  of  the  Sophomore  year. 
47t  Co-ordinate  with  Latin  and  Greek. 


326  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

What  is  the  relative  prominence  of  matliematics  in  your  course  of  study  as  shown  hy  hours 
per  iveelc  and  per  year  ? — Continued. 

48.  About  one  hour  out  of  every  four. 

49.  Five  hours  a  week  for  two  years  and.  one  year  additional,  which  is  elective. 

50.  About  the  same. 

51.  As  great  as  that  of  any  other  subject. 

52.  Equal  with  language  and  science,  until  Junior  year. 

53.  Has  more  time  than  any  other  study. 

55.  Five  out  of  fifteen  per  week  for  three  years  of  the  course.  No  mathe- 
matics in  the  last  year  of  the  course. 

56.  It  receives  more  time  than  any  other  subject  taught. 

57.  First. 

59.  About  two-fifths  of  the  time  in  the  various  schools. 

60.  As  about  five  to  four  in  comparison  with  language  and  natural  science. 

61.  First. 

62.  It  rants  with  any  other  study  in  prominence. 

63.  It  is  desired  to  make  it  equally  prominent  with  other  subjects. 

64.  One  hour  daily  in  class-room. 

65.  Four  hours  per  week  are  devoted  to  mathematics  throughout  the  course. 
It  receives  about  equal  attention  with  any  other  subject. 

66.  Quite  as  prominent  as  classics. 

67.  Under  our  "  group"  system,  under-graduate  students  who  include  in  their 
"group"  of  studies  a  minor  course  in  mathematics  devote  one-third  of  their 
time  for  one  year  (as  measured  by  hours  per  week)  to  mathematics ;  those  who 
take  a  major  course  in  mathematics  devote  to  it  one-third  of  their  time  for  two 
years.  The  whole  time  of  an  under-graduate  course  is  three  years.  A  student 
need  not  include  any  mathematics  in  his  group.  Our  entrance  requirements 
include  trigonometry  and  some  analytical  geometry. 

68.  About  one-fourth  of  required  time  is  devoted  to  pure  mathematics. 

69.  Stands  near  bottom  of  the  list. 

70.  Classics,  science,  and  mathematics  have  equal  prominence. 

71.  During  the  first  year  for  all  students,  thirty -three  and  one-third  per  cent, 
of  recitation  periods  is  for  mathematics.  The  time  for  preparation  would  be 
larger.  During  the  second  and  third  years  the  engineers  gave  about  twenty-five 
per  cent,  to  pure  and  twenty-five  per  cent,  to  applied  mathematics.  During 
the  Senior  year  about  twenty-five  per  cent,  to  applied  mathematics.  Other 
students  give  but  little  time  to  mathematics  after  the  first  year. 

72.  Four  hours  out  of  fifteen  per  week  in  Freshman  year. 

73.  Thirty-three  and  one-third  per  cent. 

74.  Freshman  year  four  hours  per  week,  i.  e.,  twenty-five  per  cent,  is  required 
throughout  the  year.  Four  hours  per  week  elective  is  ofli"ered  in  Sophomore 
and  Junior  years. 

75.  Five  hours  per  week  for  the  first  two  years  of  the  course. 

76.  It  stands  third  in  the  course. 

77.  Less  prominent  than  the  classics,  except  in  the  academy  and  in  the  in- 
ductive science  courses. 

78.  It  stands  on  the  same  level  with  Latin  and  Greek — our  courses  beiug  (like 
those  of  Harvard  College)  elective. 

79.  It  is  on  an  equality  with  Greek. 

fiO.  Algebra,  plane  geometry,  plane  trigonometry  are  required,  five  recitations 
per  week  during  Freshman  and  Sophomore  years.  Mathematics  is  elective  three 
hours  per  week  during  reet  of  course. 

82.  Five  hours  per  week  for  thirty-eight  weeks  per  year,  or  nearly  one-third 
of  the  whole  work. 

84.  In  first  year,  one-fourth  the  time;  second  and  third,  one-fifth ;  none  in  the 
last. 


I 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.    327 

What  is  the  j'elaiive  prominence  of  mathematics  in  your  course  of  study  as  shown  iy  hours 
per  weeh  and  per  year  ? — Continued. 

85.  One-tliird  of  the  time. 

86.  Of  required  work,  mathematics  has  about  fifteen  per  cent,  out  of  the  fifty- 
two  hours  weekly. 

87.  Our  students  average  in  three  years  six  and  one-third  hours  per  week  of 
mathematics,  to  four  and  one-half  of  language,  to  four  of  physics  and  chemistry, 
mineralogy  and  geology  altogether. 

88.  It  is  dijfferent  in  the  different  years.  For  the  four  years  it  is  ahout  one  to 
ten. 

89.  One  to  six. 

90.  About  par. 

91.  The  hours  are  about  equal,  taken  as  a  whole.  For  the  degree  of  B.  S. 
they  far  exceed. 

92.  A  high  importance ;  fifteen  terms  (including  the  preparatory  course) ; 
five  hours  per  week,  forty  weeks  per  year,  for  five  years. 

93.  Sub-Freshman  year,  two-sevenths  of  entire  work;  Freshman  year  two- 
ninths  ;  Junior  year(elective)  one-fourth;  Senior  year  (elective)  one-sixth.  This 
does  not  include  mechanics,  surveying,  and  other  applied  mathematics. 

95.  Four  hours  per  week ;  most  other  studies  (non-professional),  three. 

96.  Daily  recitations,  one-half  hour  each. 

97.  In  the  scientific  courses  it  is  first ;  in  the  classical  and  literary,  second 
(Latin  and  Greek,  respectively,  English  first). 

98.  Mathematics  is  the  ground  work  of  the  institution,  preparatory  and  coin- 
cident with  the  courses  in  engineering. 

99.  Two  hours  daily  devoted  to  mathematics  during  the  session  of  ten 
months. 

100.  Foorteen-sixtieths  of  the  four  years'  coxirse. 

101.  Twenty-five  per  cent. 

102.  About  one-third  of  the  whole  time ;  ten  hours  per  week,  480  hours  per 
year. 

103.  Each  class  averages  five  hours  per  week,  per  year. 

104.  Same,  no  prominence. 

105.  In  the  preparatory  course,  total  hours  per  week  fifteen ;  mathematical 
average,  three  and  one-third.  In  college  classes,  total  hours  per  week,  fifteen ; 
mathematical  average,  two  and  eleven-twelfths. 

106.  The  mathematical  course  for  the  majority  of  our  courses  is  completed  the 
Freshman  year,  having  five-sixteenths  of  the  time. 

107.  Leading  study. 

108.  Five  twenty-fourths  of  the  whole. 

109.  Mathematics  ranks  with  Latin  and  Greek  throughout,  each  getting  four-- 
fifteenths  of  the  time  in  first  two  years ;   elective  in  third. 

110.  Until  this  year,  more  than  half  the  time  was  given  to  mathematics ;  now, 
perhaps,  one-third. 

111.  We  require  two  and  one-half  hours  per  week. 

112.  Considering  the  hours  devoted  to  mathematics,  it  ranks  with  any  other 
subject. 

113.  Mathematics  and  Latin  have  each  three  hours  a  day. 

114.  Freshman  year,  one- third ;  Sophomore  year,  four-fifteenths ;  Junior  and 
Senior  years,  optional. 

115.  The  courses  in  my  department  are  elective.  Question  cannot  well  be 
answered. 

116.  Five  hours  per  week  out  of  fifteen  in  Freshman  year ;  three  hours  per 
week  out  of  fifteen,  Sophomore  year  for  classical ;  four  out  of  fifteen  for  eoien- 
tifio. 


328  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

What  is  the  relative  ^prominence  of  mathematics  in  your  course  of  study  as  shown  hy  hours 
per  iveek  andjom'  year  ? — Continued. 

117.  Freshman  year,  five  hours  per  week  out  of  sisteen ;  Sophomore,  five  out 
of  seventeen  in  first  terra,  and  three  out  of  eighteen,  second  term. 

118.  In  scientific  course  sisteen  one-hundredths  of  whole  time,  or  twenty- 
eight  one-hundred ths,  including  descriptive  geometry  and  mechanics  and  as- 
tronomy. In  classical  course  thirteen  one-hundredths  and  sixteen  one-hun- 
dredths. 

119.  Four  hours  per  week  required  during  the  entire  Freshman  dnd  Sopho- 
more years,  and  the  second  term  of  the  Junior  year.  From  two  to  sis  hours  per 
■week  may  be  elective  during  the  other  terms. 

120.  During  the  first  year,  time  devoted  to  mathematics  is  to  time  devoted  to 
modern  languages  as  four  and  one-half  is  to  three.  During  the  second  year, 
time  devoted  to  mathematics  is  about  the  same  as  the  time  devoted  to  languages 
and  drawing. 

123.  It  is  on  an  equality  with  the  subjects  taught  in  the  other  departments. 

124.  Mathematics  is  one  of  our  most  important  subjects.  Three  professors 
give  jointly  forty-five  hours  per  week  to  mathematical  instruction,  for  twenty- 
eix  weeks,  and  thirty-nine  hours  per  week  the  remaining  eleven  weeks  of  the 
session. 

125.  Freshmen  five-fifteenths,  Sophomores  three-fifteenths,  Juniors  (elective) 
two-fifteenths.  Seniors  (elective)  two-fifteenths. 

126.  It  occupies  about  one  -third  of  the  whole  course. 

127.  Full  work  in  all  subjects  fifteen  hours  per  week.  In  mathematics  five 
hours  per  week  whenever  any  mathematical  subject  is  studied.  In  the  prepar- 
atory department  algebra  is  required  in  all  the  courses  three  terms  (one  full 
year),  and  in  the  science  courses  four  terms.  Plane  geometry  is  required 
through  the  last  term  of  the  Senior  preparatory  year.  Then,  in  the  college 
course,  we  have  solid  geometry,  trigonometry,  surveying,  analytical  geometry, 
and  calculus,  one  term  each.  A  second  term  of  analytical  geometry  and  cal- 
culus is  required  of  the  scientific  students ;  and  in  the  philosophical  course, 
students  elect  between  a  second  term  of  calculus  and  practical  chemistry. 

128.  All  courses  are  four  or  five  hours  a  week.  In  classical  course,  mathe- 
matics have  378  hours,  required  and  elective,  out  of  a  total  of  4,077  hours. 

129.  Two  hundred  and  seventy  hours  distributed  through  two  years  with 
opportunity  of  election  in  addition. 

130.  Four  hours  out  of  fifteen  per  week  during  two  years  for  all  the  class ; 
then  four  out  of  fifteen  during  another  year  for  electives. 

131.  One-fourth  up  to  second  term  of  Sophomore  year ;  from  that  point  all 
subjects  in  the  course  are  elective. 

132.  We  have  too  many  courses  to  make  any  general  statement. 

133.  About  the  same  as  ancient  languages. 

134.  Greater  than  others,  except  English. 

135.  Freshman  year,  five  hours  out  of  eighteen  per  week. 

136.  Mathematics  is  on  an  equality  with  all  other  courses. 

137.  A  little  more  than  one-fourth  of  the  student's  time  is  given  to  mathe- 
matics in  Freshman  and  Sophomore  years ;  a  little  less  than  one-fourth  during 
the  remainder  of  the  course. 

139.  Full  course,  four  hours  per  week. 

140.  About  twenty-five  per  cent,  of  total. 

141.  On  a  par  with  the  classical. 

142.  First  in  the  course. 

143.  Pure  mathematics  thirteen  and  sis-tenths  per  cent,  of  required  work. 
It  may  be  thirty-four  per  cent,  of  elective  work.  It  may  be  one-fifth  of  the 
whole  course. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.    329 

What  is  the  relative  prominence  of  niaihematics  in  your  course  of  study  as  sUoivn  by  hours 
per  loeeTc  and  per  year? — Continued. 

144.  First  year  about  thirty- three  and  one-third  per  cent,  of  time  to  mathe- 
matios;  second  year,  fifteen  per  cent.;  third  year,  twenty-five  per  cent. ;  fourth 
year,  fifteen  per  cent. 

145.  Out  of  the  required  eighteen  hours  per  weei:,  literary  students  get  in 
the  first  year  five,  second  year  three,  third  year  nought,  fourth  year  one  and  one- 
half;  total,  nine  and  one-half  out  of  seventy-two ;  scientific  students  get  in  the 
first  year  five,  second  year  four  and  one-half,  third  year  four  and  one-half,  fourth 
year  one  and  one-half;  total,  fifteen  and  one-half  out  of  seventy-two. 

146.  We  regard  it  of  greatest  importance. 

147.  In  the  preparatory  course  one-third  of  the  study  is  mathematical,  i  e. 
185  hours  a  year  out  of  555.  The  same  in  the  Freshman  year.  In  the  Sopho- 
more year  135  hours  out  of  555.    After  that,  none. 

148.  Twenty-five  per  cent,  of  the  student's  time  is  devoted  to  mathematics 
until  he  completes  the  Sophomore  year.  Besides,  the  students  in  engineering 
devote  twelve  and  one-half  per  cent,  of  their  time  in  Junior  year  to  mathematics. 

149.  About  one-fourth  the  time  is  devoted  to  mathematics. 

150.  Mathematics  to  science  about  equal ;  mathematics  to  language  about 
four  to  one. 

151.  About  the  same  time  is  given  to  mathematics  as  to  other  branches,  viz> 
five  recitations  per  week,  except  in  last  year,  three  times. 

152.  Mathematics,  Latin,  and  Greek  have  each  four  hours  per  week  for 
Freshman  and  Sophomore  years ;  no  other  subjects  have  as  much  time.  After 
Sophomore  year  mathematics  is  elective. 

153.  More  prominent  than  any  other  subject  except  English  and  equal  to 
that. 

154.  It  stands  first. 

155.  It  is  as  prominent  as  any  other  branch  of  study.  The  Junior  class,  witli 
whichi  the  college  work  properly  begins,  has  five  recitations  per  week,  each 
one  hour  long.  The  intermediate  class  has  four  per  week,  the  Senior  has  thr<^/e 
per  week,  and  in  applied  mathematics  there  are  three  per  week. 

156.  About  twenty  hours  per  week,  or  eight  hundred  hours  per  session. 

157.  It  is  given  as  much  time  as  other  subjects,  five  hours  per  week  in  the 
Freshman  year,  five  hours  per  week  in  the  Sophomore  year,  three  hours  per 
week  in  the  Junior  year,  and  two  hours  per  week  in  the  Senior  year. 

158.  Our  system  of  independent  schools  and  free  elective  courses  enables  us 
to  give  a  positive  statement  that,  as  a  rule  (with  a  few  exceptional  years),  the 
school  of  mathematics  is  the  most  largely  attended  school  in  the  academic  de- 
partment. The  number  of  lecture  hours  per  week  for  under-graduates  is  thir- 
teen. 

159.  Mathematics  occupies  a  more  prominent  position  in  our  schedule  than 
any  other  branch. 

160.  First. 

161.  It  occupies  about  one-third  of  the  time  devoted  to  the  course  of  instruc» 
tion. 

162.  Five  sections  per  week  or  nearly  one- third  of  time  for  first  two  years. 

163.  It  is  probably  on  about  the  same  footing  as  the  other  chief  branches  of 
study. 

164.  Our  course  in  mathematics  is  very  prominent,  requiring  one-third  the 
student's  time  through  the  Sophomore  year. 

165.  Three  to  two. 
M6.  One-half. 

167.  Classical  course  :  Freshman  year,  one-third  of  time  required ;  Sophomore 
year,  one-third  elective;  Junior  year,  one-ninth  elective.  Scientific  course: 
Freshman  year,   one-third  required;   Sophomore  year,  two-ninths  required, 


330  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

What  is  the  relative  prominence  of  mathemaiica  in  your  course  of  study  as  shown  hy  houra 
per  iveelc  and  per  year  ? — Continued, 
one-nintli  elective;  Junior  year,  ono-ninth  required;  Senior  year,  one-ninth 
elective. 

168.  Almost  the  least  prominent  thing  in  the  course,  as  ours  is  classical,  with 
a  leaning  to  natural  sciences. 

(o)  Do  you  favor  memorizing  rules  in  ailgebr  a  f   (J))  What  reforms  are  needed  in  teaching 

the  same  f 
'  1.  (a)  No. 

2.  (a)  No.  (J)  It  ought  not  to  he  taught  to  such  young  hoys,  who  contract 
the  incurahle  habit  of  learning  it  by  rote. 

3.  Principles,  but  not  rules. 

4.  (a)  No.  (&)  Eules  and  principles  should  be  deduced  from  examples ;  a 
more  thorough  drill  in  algebraic  language,  especially  in  the  meaning  and  use 
of  signs,  exponents,  etc. 

5.  Do  not  use  test-book  too  closely. 

6.  (a)  Yes.    (&)  More  practical  application  should  be  given. 

7.  No. 

8.  No. 

9.  (a)  I  prefer  formulas.  (J)  More  thoroughness  and  better  understanding 
of  elementary  principles,  with  reviews  and  drilling. 

10.  (a)  No.  (6)  Algebra  should  be  taught  just  as  arithmetic,  wholly  by  the 
analytic  method. 

11.  (a)  Some  of  the  rules.     (&)  I  do  not  know. 

12.  (a)  To  a  limited  extent.  (6)  More  of  the  spirit  and  reason  and  less  mere 
mechanical  solution. 

13.  (a)  No.  (6)  The  modern  methods,  as  determinants,  etc.,  should  be  intro- 
duced as  soon  as  possible. 

14.  (a)  Yes. 

15.  (a)  No.  (6)  A  larger  number  of  simple  problems ;  a  less  number  of  diffi- 
cult demonstrations,  such  as  those  in  logarithms,  the  binomial  formula,  etc.; 
an  earlier  introduction  to  the  methods  and  notation  of  the  calculus. 

16.  (a)  No. 

17.  (a)  Very  little.  (&)  Anything  which  will  make  it  less  a  collection  of.dry 
bones,  and  more  a  living  and  beautiful  science. 

18.  (a)  Yes. 

19.  (b)  By  proper  classification  the  number  of  propositions  could  be  materi- 
ally reduced  and  the  number  of  important  theorems  and  constructions  for  origi- 
nal work  could  be  materially  increased.  (Professor  Gordon),  (o)  No.  (6) 
More  attention  should  be  paid  to  explaining  and  illustrating  the  principles  in- 
volved in  operations,  and  the  embodying  of  questions  to  test  the  understanding 
of  those  principles ;  e.  g.,  tvhy  x  —  &  =  10  is  equal  to  a;  =  10  -f-  &/  v}hy  does  + 
X  —  =  —  ?  etc.     (Professor  Draper.) 

20.  (a)  Yes.  (&)  That  the  sense  of  the  rules  shall  he  known  when  the  mem- 
orizing is  complete. 

21.  (a)  No.  (&)  Teaching  needs  to  be  less  mechanical.  The  reasons  for  proc- 
esses need  to  be  taught  more. 

22.  No ;  the  inductive  method  should  be  used  first. 

23.  (a)  Yes. 

24.  No  ;  thorough  drill  in  substituting  numerical  quantities  for  literal. 

25.  (c)  I  do  not.  (b)  In  teaching  the  elements  as  few  formal  demonstrations 
as  possible  should  be  used — first  a  working  knowledge,  and  then  philosophize. 

26.  (a)  Do  not.     (6)  Practical  examples. 

27.  No ;  more  attention  to  fundamental  principles,  clear  teaching  why  signs 
ate  changed  in  transposition,  etc. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.    331 

(a)  Do  you  favor  memorising  rules  in  algebra?    (6)  What  reforms  are  needed  in  teaching 

the  same  ? — Continued. 

28.  No ;  more  prominence  to  principles  and  less  of  method.  Students  should 
do  more  private  work. 

29.  No;  it  should  be  freed  from  its  mechanical  character.  Algebra  should  be 
seen. 

30.  Not  much  ;  less  memorizing,  more  analysis,  more  thoroughness. 

31.  We  require  the  principles  involved,  rather  than  the  exact  words  of  a  rule. 

32.  Yes;  only  the  most  important  rules  and  theorems  should  be  memorized, 
but  those  thoroughly. 

33.  Emphatically  no;  more  principle  and  why;  less  toughing,  disgusting 
gymnastics. 

34.  No. 

35.  No;  more  familiarity  with  technique  ;  less  mechanism. 

36.  No. 

37.  No,  sir. 

38.  (a)  Not  in  general.  {I)  In  general,  I  should  say  a  more  thorough  teach- 
ing of  the  principle  and  reasoning  of  algebra. 

39.  (a)  Yes.     (6)  More  ought  to  be  taught. 

40.  (a)  Students  are  urged  to  state  operations  and  principles  clearly  and 
briefly  without  much  regard  to  the  text,  {h)  Many  problems  (original  and 
otherwise)  should  be  solved  mentally. 

41.  (a)  No.  (6)  More  careful  attention  to  the  interpretation  of  literal  equa- 
tions. 

42.  (a)  To  some  extent. 

43.  (a)  No. 

44.  (a)  I  do  not.  {I)  More  drill  on  simple  exercises,  and  fewer  difficult 
problems. 

45.  (a)  I  do  not.  (&)  Such  as  will  render  the  mind  able  to  deal  with  the 
principles  in  forming  rules. 

46.  (fl)  Some  of  them.  (&)  Too  little  time  seems  to  be  given  to  the  study 
of  algebra. 

47.  (a)  Not  verbatim.  (J>)  More  rigorous  proofs ;  more  noting  of  analogies ; 
more  as  a  preparation  for  higher  work  than  the  solving  of  problems  as  mere 
puzzles. 

48.  (a)  Yes. 

49.  (a)  No. 

50.  (a)  I  do. 

51.  (a)  No.  (J)  Methods  are  learned  by  practice,  and  rules  evolved  there- 
from. 

52.  (a)  No. 

53.  (a)  Very  few.  (6)  There  should  be  more  practical  application  of  its 
principles. 

54.  (a)  Not  mechanically,  (l)  The  reform  of  good  common  sense,  and  clear, 
simple  presentation. 

55.  (a)  No. 

56.  (a)  No.  (&)  Keep  students  out  of  it  until  they  have  passed  the  discus- 
sion of  arithmetic. 

57.  (a)  Yes.  (6)  We  need  a  simpler  and  at  the  same  time  fuller  elementary 
book. 

58.  (a)  At  the  beginning.    (&)  More  familiarity  with  principles. 

59.  (a)  Not  generally,  (b)  The  teacher  should  assist  the  pupil  to  make  his 
own  rules. 

60.  (a)  No.  (6)  A  correct  reading  of  algebraic  expressions  in  algebraic  lan- 
guage, and  a  clear  analysis  of  work  done. 


332  TEACHING   AND   HISTOEY    OF   MATHEMATICS. 

(a)  Do  you  favor  memonsing  rules  in  algebra  ?    (h)  What  reforms  are  needed  in  teaching 

the  same  ?---Contmued. 
61.  (a)  I  do  not.    (h)  Eequire  pupils  to  think  and  not  to  be  machines  or  juga 
to  1)6  filled. 

G2.  (a)  To  a  certain  extent.  (&)  There  is  need  of  impressing  the  students  in 
some  way  with  the  idea  of  the  practical  value  of  the  study  and  of  creating  an 
interest  in  it. 

63.  (a)  No.  (5)  Teachers  should  wait  till  their  pupils_^are  prepared  to  begin 
the  study.    It  should  be  thoroughly  taught. 

64.  (a)  Yes. 

65.  (a)  Yes.  (5)  The  subject  ought  to  be  presented  freer  from  technicalities 
than  text-books  give  it.    Unnecessary  parts  ought  to  be  left  out. 

66.  (a)  Yes. 

68.  (ffl)No. 

69.  (a)  No.  (&)  Drill  on  the  principles  and  raison  d'etre  for  formula. 

70.  (a)  Yes ;  when  once  thoroughly  understood. 

71.  (a)  No.  (6)  The  use  and  meaning  of  exponents  and  of  the  negative  sign 
are  not  made  as  clear  as  they  should  be.    More  accuracy. 

72.  (a)  No. 

73.  (a)  No.  (5)  Less  rules  and  more  thinking.  The  less  memorizing  in 
mathematics,  the  better  the  results. 

74.  (a)  No. 

75.  (a)  Yes. 

76.  (a)  No.  (h)  The  teacher  should  lead  with  the  general  demonstration  of 
each  subject  in  form  of  lectures. 

77.  (a)  No.  (h)  Pupils  should  be  required  more  generally  to  demonstrate 
principles  and  work  from  them  rather  than  from  rules  and  fornml;©. 

78.  (a)  We  do  not  teach  elementary  geometry. 

79.  (a)  Yea.  (&)  The  rules  should  be  proved  as  strictly  as  any  proposition  in 
geometry. 

80.  (a)  No.     (h)  More  classification  of  subjects. 

82.  (a)  No.  (h)  The  chief  cause  of  failure  in  many  cases  is  not  doing  enough 
miscellaneous  examples  for  practice. 

83.  (a)  Yes. 

84.  (a)  Some.     (&)  None. 

85.  (a)  A  more  logical  arrangement  of  the  different  sections  of  the  subject; 
more  exami^les,  and  so  given  as  to  form  a  constant  review  of  the  ground  already 
gone  over ;  application  of  business  methods  to  the  revision  of  many  rules  and 
methods. 

86.  (a)  No.  (h)  More  thoroughness,  practicality,  and  solidity  of  teaching — 
the  German  system. 

87.  (a)  To  some  extent.  (&)  For  a  course  of  study  like  ours  I  think  more 
emphasis  should  be  put  on  thoroughness  than  extent  of  ground  covered. 

88.  (a)  Yes. 

89.  (a)  No. 

90.  (a)  No.     (b)  More  attention  to  reasoning  processes. 

91.  (a)  I  do  not.  (&)  More  independence  of  books  and  greater  original  inves- 
tigation. 

92.  (o)  No.  (&)  Less  memorizing  and  more  thinking,  both  on  the  part  of 
teacher  and  student. 

93.  (a)  Yes.  (b)  Pupils  should  be  taught  to  state  a  proposition  and  follow  it 
with  a  general  demonstration,  as  in  geometry. 

94.  (a)  Yes — No!  Teach  the  pupil  to  develop  the  principle,  and  to  formulate 
his  own  rule  for  it  and  for  his  process. 

95.  (a)  Not  much.    (&)  Omit  attempts  to  exhaust  each  subject  as  it  comes  up. 

96.  (a)  No. 


MATHEMATICAL    TEACHING   AT   THE    PRESENT   TIME.        333 

(O;  JJO  you  favor  memorising  rules  in  algebra  ?    (&)  What  reforms  are  needed  in  teaching 

the  same? — Continued. 

97.  (a)  No.  (6)  Blind,  unreasoning,  mechanical  solution  of  equations  needs 
abatement;  the  doctrines  of  forms  and  series,  advancement. 

98.  (a)  No!!  Battle  the  bones  of  the  algebraic  skeleton,  as  exhibited  generally  in 
this  country,  and  shoxo  it  in  its  living,  breathing  continuity  and  beauty  of  FORM, 
Give  a  conception  of  the  magnificent  power  of  analysis. 

99.  (a)  No.     (b)  Much  desired  in  test-books,  at  least  many  of  them. 

100.  (a)  Not  verbatim. 

101.  (a)  No.  '  - 

102.  (a)  No.     (6)  More  attention  to  principles  and  less  to  problems. 

103.  (a)  For  immature  students,  yes.  (&)  The  method  of  teaching  must,  I 
think,  vary  under  different  circumstances.  The  principle  idea  should  be  to  pre- 
vent the  student  thinlcing  it  difficult. 

104.  (a)  No.    (&)  Knowledge,  generally. 

105.  La)  No. 

106.  C6)  I  favor  thorough  mastering  of  the  reasoning  used  in  deducing  formulae, 
also  memorizing  for  ready  use. 

107.  (a)  Not  word  for  word.  (&)  Digesting  subject  as  a  whole,  especially  on 
review. 

108.  (rt)  No. 

109.  (ft)  No,  with  few  exceptions.  (6)  A  more  thorough  treatment  of  a 
smaller  number  of  subjects  ;  use  of  determinants,  less  fractions. 

110.  (ft)  No.  (6)  More  stress  should  be  laid  on  factoring,  less  on  the  the- 
ory— more  of  the  solid  work  with  a  broader  view  of  its  application. 

111.  (a)  We  do  not.     (&)  More  mental  exercise  and  less  blackboard  work. 

112.  (a)  I  do  not.  (6)  More  attention  should  be  paid  to  generalization  than 
it  usually  receives. 

113.  (a)  No. 

114.  (ft)  No. 

115.  (ft)  That  depends,  (b)  We  need  no  reforms  (Professor  Peck),  (a)  Yes. 
(&)  In  the  preparatory  schools  more  work  should  bo  done  independently  of  the 

'    text-book,  and  a  more  elaborate  elucidation  of  fundamental  pj'inciples  should 
be  given  (Tutor  Fiske). 

116.  (a)  No.  (&)  In  general,  greater  attention  to  accuracy;  in  particular, 
more  attention  to  theory  of  exponents  and  radicals. 

117.  (a)  Yes ;  either  those  of  the  text-book  or  carefully  prepared  ones.  More 
"  why  "  needed. 

118.  (a)  Yes  ;  so  far'as  to  secure  accuracy  of  expression  and  as  a  mode  of  fix- 
ing methods  clearly  in  the  mind. 

119.  (ft)  Not  to  a  large  extent.  (6)  I  think  the  student  should  be  taught  to 
rely  upon  his  logical  powers,  rather  than  his  memory. 

120.  (ft)  No.  {b)  Methods  that  develop  a  clear  understanding  of  each  proc- 
ess and  ability  to  explain  clearly,  in  place  of  a  knowledge  of  rules  without 
understanding. 

121.  (ft)  No.  (Z>)  More  of  the  inductive  method ;  and  the  abolition  of  much 
that  may  be  interesting  theoretically,  but  of  little  practical  use. 

122.  (ft)  In  very  few  cases. 

123.  (ft)  Yes.  (&)  An  improvement  in  the  speed  with  which  the  mechanical 
processes  are  done. 

124.  (ft)  Yes  ;  saves  time.     (6)  Get  teachers  who  know  more. 

125.  (ft)  Hardly.  (&)  1.  Opposite  numbers  ought  to  be  given  a  full  treatment, 
including  all  the  rules  for  signs,  with  illustrations  and  a  considerable  number 
of  examples  and  problems  in  their  use,  before  the  literal  notation  is  begun.  2.  In 
the  former  the  reason  for  the  use  of  -{-  and  —  to  mark  the  series  ought  to  be 


334  TEACHING   AND    HISTOEY    OF   MATHEMATICS. 

^^r^  T)o  you  favor  memorising  rules  in  algebra  t    (b)  What  reforms  are  needed  in  teaching 

the  same  ? — Continued, 
brotigM  out  simply  and  plainly,  and  justified.  3.  The  fact  that  in  element- 
ary algebra  the  letters  always  stand  for  numbers  ought  to  be  reiterated  to 
avoid  obscurity  of  ideas  in  the  learner's  mind.  4.  The  treatment  of  the  equa- 
tion should  be  analogous  to  that  employed  in  geometry.  The  method  of  writing 
references  to  axioms,  etc.,  at  the  right  of  the  page,  familiar  to  those  who  have 
used  Wentworth's  Geometry,  can  be  employed  in  algebra  to  even  greater  ad- 
vantage. 

126.  (a)  No,     ■ 

127.  {a)  Some  of  them,  (b)  Examinations  of  students  from  many  places  con- 
vince me  that  algebra  should  be  taught  more  thoroughly  than  it  is  in  most  of 
the  schools. 

128.  (o)  No.  (b)  Explanation  to  be  really  so,  and  work  done  at  time  of  ex- 
planation as  far  as  possible.  Many  comparatively  simple  problems,  not  puz- 
zles. New  work  in  hour.  Students  to  be  ranked  according  to  actual  work 
done  in  problems.    Much  board  work  by  entire  class. 

129.  (a)  Yes  and  no.  (b)  Better  elementary  text-books,  better  preparation 
on  part  of  teacher  ;  more  rigid  demonstrations  of  the  principles  of  a  science. 

130.  (a)  Not  in  general,  (b)  More  attention  should  be  given  to  the  axioms 
and  the  fundamental  laws  and  their  connection  with  the  subject,  and  more  at- 
tention to  the  theory  of  simultaneous  equations. 

131.  (a)  Y'es,  most  important,  but  not  necessarily  in  the  words  of  text.  (6) 
In  the  larger  colleges  algebra  is  mostly  taught  by  tutors,  who  hold  temporary 
appointments,  and  do  not  expect  to  make  teaching  their  life  work.  Algebra 
as  well  as  calculus  should  be  taught  by  a  permanent  professor. 

133.  (a)  Yes,  but  not  in  rigorous  form,  (ft)  Greater  facility  in  their  use  with 
a  more  intelligent  understanding  of  them. 

134.  (a)  Yes,  but  they  must  also  be  thoroughly  understood. 

135.  (a)  Yes,  for  average  student,  (b)  Examples  given  should  be  made  more 
modern  and  practical.  The  theory  of  functions  should  be  incorporated,  beginning 
with  simple  elements.  This  will  malce  the  whole  subject  of  series,  etc.,  easy  for  the 
student. 

136.  (a)  Very  few. 

137.  (a)  Yes.  (&)  Let  us  have  live,  enthusiastic,  and  competent  teachers — 
Buch  as  will  teach  the  subject  rather  than  the  text-book. 

138.  (a)  The  more  important,  yes.  (&)  For  the  preparatory  work,  greater 
thoroughness  is  much  needed. 

139.  (a)  No,     (5)  Explain  by  common  sense  and  not  by  rule, 

140.  (a)  It  does  no  harm.    (Z>)  The  current  text-books  are  too  arithmetical. 

141.  (a)  No. 

142.  (a)  No. 

143.  (a)  No.  (&)  Less  mechanical  work;  more  thought.  Students  should 
be  taught  to  think !  think! !  think ! ! ! 

144.  (a)  No.  (6)  With  such  text-books  as  Hall  and  Knight's  Elementary 
Algebra  and  same  as  C.  A.  Smith's  or  Todhunter's  Higher  Algebra ;  no  reform 
needed. 

145.  (a)  No.  (&)  Greater  attention  to  menial  and  inventional  algebra,  and  to 
numerical  and  geometrical  applications  and  illustrations. 

146.  (a)  No. 

147.  (a)  I  recommend  the  memorizing  of  the  rules,  unless  the  pupils  furnish 
a  good  working  rule  of  their  own  (a  rare  case). 

148.  (a)  No.  (b)  A  more  thorough  drill  in  factoring  and  in  fractions,  and  in 
putting  into  words  the  ideas  conveyed  by  its  symbols,  equations,  and  opera- 
tions.   Also  greater  precision  of  expression. 


MATHEMATICAL    TEACHING   AT    THE    PEESENT   TIME.        335 

(a)  Do  you  favor  memorizing  rules  in  algebra  ?    (i)  What  reforms  are  needed  in  teaching 

the  same  ? — Continued. 

149.  (a)  I  do  not. 

150.  (a)  No. 

151.  (a)  I  do  not.  (&)  The  pupils  learn  to  do  by  doing.  Hence,  instead  of 
having  pupils  waste  their  time  on  abstract  demonstrations,  let  them  solve  nu- 
merous problems  of  every  variety.     It  is  only  practice  that  makes  perfect. 

152.  (a)  No. 

153.  (a)  No.    (J)  The  founding  of  all  algebra  upon  the  laws  of  operation. 

154.  (o)  No. 

155.  (a)  Yes. 

156.  (a)  Not  at  all.  (&)  The  pupil  should  be  taught  to  think  rather  than  to 
work  hy  rule.    More  thoroughness  needed. 

;    157.  (a)  I  do  not.     (&)  I  think  that  the  student  should  be  required  to  cou- 
Btruct  his  own  rules  as  far  as  possible. 

159.  (a)  But  few.  (6)  Principles  are  apt  to  be  lost  sight  of  in  the  strict  and 
close  adherence  to  rules. 

160.  (a)  No.  (i)  More  thorough  drill  is  needed,  especially  in  the  elementary 
principles. 

-     161.  (o)  Yes.     (&)  Broader  views  of  algebraic  operations ;  more  generalizing 
and  greater  exactness  of  language. 

162.  (a)  Yes. 

163.  (a)  Yes.     (6)  The  rules  should  be  demonstrated  oftener  than  they  are. 

164.  (a)  No.     (&)  To  develop  the  subject  by  original  investigation. 

166.  (o)  Yes. 

167.  (a)  To  but  very  slight  extent.  (&)  Less  formality  and  more  "  realism ;" 
introduction  of  principles  often  held  back  until  higher  branches  are  reached, 
€.  g.,  factors  of  direction,  differentials,  etc. 

168.  (a)  Only  very  few.  (&)  More  attention  to  problems  involving  principles 
and  less  to  puzzles. 

(a)  To  what  extent  are  models  used  in  geometry  f  (6)  To  what  extent  and  with  what  success 
original  exercises  f  (o)  Do  you  favor  memorizing  veriatim  the  theorems  (not  the  demon- 
strations) in  geometry  ?    What  reforms  are  needed  in  teaching  the  same  ? 

1.  (a)  Class-room  very  poorly  supplied,  but  we  use  the  few  we  do  possess  as 
much  as  possible,  (b)  Such  exercises  are  given  every  day  and  are  found  to  be 
very  beneficial,     (c)  No. 

2.  (a)  Moderately,  to  explain  effects  of  perspective  on  the  black-board.  (6) 
To  a  very  moderate  extent  with  the  great  majority  of  students,  to  a  great  extent 
with  the  best. 

3.  (a)  When  models  have  been  used  it  has  facilitated  the  work,    (c)  Yes. 

4.  (a)  They  are  used  to  a  limited  extent.  I  question  very  much  the  advan- 
tages of  using  models,  except  with  beginners,  or  rather  with  those  who  are 
studying  works  introductory  to  regular  demonstration.  (6)  To  a  limited  ex- 
tent, (c)  Yes;  more  original  work ;  more  attention  to  logical  processes,  clear- 
ness and  accuracy  of  statement.  I  change  the  figures,  i.  e.,  their  relative  posi- 
tion, so  that  the  demonstration  shall  be  reasoning  and  not  memory. 

5.  (a)  Use  them  to  a  great  extent.  (J»)  Original  exercises  with  fine  success. 
(c)  No ;  some. 

6.  (a)  The  text-book  quite  closely  followed.  (&)  Some  daily  and  with  good 
success,    (c)  Yes. 

7.  (a)  Not  used.    (&)  Used  to  some  extent,    (c)  No. 

8.  (a)  To  a  large  extent,  (b)  One-fifth  of  work  in  geometry  ia  in  original  es- 
eioise ;  the  success  is  good,    (c)  Yea. 


336  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

(a)  To  rvhat  extent  are  models  usedin  geometry  ?  (5)  To  xoliat  extent  and  ivith  what  success 
original  exercises?  (c)  Do  you  favor  memorizing  verbatim  the  theorems  {not  the  demon- 
strations) in  geometry  ?     What  reforms  are  needed  in  teaching  the  same  ? — Continued. 

9.  (a)  To  a  considerable  extent  in  the  lower  grades,  (i)  Very  extensively 
and  with  very  satisfactory  results,  (c)  As  a  general  thing,  I  am  of  the  opinion 
that  too  little  time  is  given  to  the  subject  io  secure  the  best  results. 

10.  (a)  Largely,  both  in  class-room  and  out-doors.  (6)  To  no  great  extent 
and  with  no  marked  success,  as  yet. 

11.  (a)  Very  little.  (&)  The  exercises  in  Welsh's  Geometry  are  used.  Some 
of  them  seem  hard  to  the  students,  but  on  the  whole  they  do  fairly  with  them, 
(c)  Almost  verbatim. 

12.  (a)  Only  moderately,  (l)  They  take  one-third  of  the  whole  time  and 
make  the  life  of  the  work,  (c)  The  rigorous  requirement  of  original  well-graded 
work  from  the  very  first. 

13.  {a)  In  descriptive  geometry  only,  (c)  It  ought  to  be  taught  more  from 
a  comparative  point  of  view. 

14.  (a)  None.  (&)  To  a  very  limited  extent  and  not  with  marked  success, 
(c)  No. 

15.  (a)  Models  are  largely  used  in  geometry  in  three  dimensions.  (6)  To  a 
small  extent  and  without  marked  success,  (c)  No;  better  drawing  in  the 
text-books,  especially  in  geometry  in  three  dimensions ;  more  attention  to  the 
drawing  of  the  studeuts  ;  less  geometry,  altogether  ;  I  think  the  importance  of 
Euclidian  geometry  as  mental  discipline  is  greatly  overestimated. 

16.  (l))  To  a  very  large  extent  and  with  excellent  success,  (c)  It  is  left  op- 
tional with  the  student. 

17.  (a)  Very  little.  (6)  Subordinate  to  a  marked  degree.  I  am  trying  to 
change  this  state  of  affairs,  (c;  Yes;  more  original  work,  also  more  compara- 
tive, not  purely  descriptive  work. 

18.  (a)  Always  used  in  teaching  solid  geometry  and  in  teaching  conic  sec- 
tions. 

19.  (a)  Forms  are  used  in  solid  geometry,  etc.,  freely,  to  aid  the  mental  con- 
ception of  the  perfect  ideals  of  mathematics  (Professor  Gordon).  But  little  used 
in  plane  geometry  (Professor  Draper).  (&)  Very  simple  exercises,  arithmetical 
application  of  geometrical  principles,  constructions,  and  problems  are  freely 
used.  Very  simple  "catch"  theorems  or  "  corollaries"  involving  some  absurd- 
ity are  occasionally  introduced  to  l)e proved!  Students  of  ordinary  intelligence 
generally  succeed  with  exercises  graduated  to  their  state  of  advancement  (Pro- 
fessor Gordon).  About  one-fifth  of  the  time  is  given  them.  Those  who  do  well 
in  the  text  and  stand  questioning  upon  it  are  fairly  successful  with  originals. 
(Professor  Draper),  (c)  Yes ;  except  in  a  few  cases  where  I  think  the  theorem 
itself  can  be  improved.  Would  begin  it  in  childhood  of  pupil ;  would  spend 
more  time  on  its  elementa,ry  principles  (Professor  Draper). 

20.  (a)  Models  are  nsed  in  solid  geometry  and  spherical  trigonometry.  (&) 
To  a  limited  extent  and  with  good  results,  (c)  Yes.  That  the  sense  of  the 
theorem  be  known  when  the  memorizing  is  complete. 

21.  (fl)  Very  little  in  plain  geometry  ;  more,  but  not  very  extensively,  in  solid 
geometry.  (&)  Used  largely  and  with  unqualified  success,  (c)  Yes.  Using 
figures  just  as  given  in  book,  using  only  propositions  already  proven,  and  many 
other  things  of  a  similar  kind  need  reformation. 

22.  (a)  To  a  limited  extent.  (&)  To  limited  extent  and  with  good  success. 
(c)  Yes,  at  first ;  it  promotes  accuracy  of  expression.  Greater  latitude  may  be 
allowed  with  advanced  studeuts. 

23.  (a)  Models  have  just  been  obtained. 

24.  (a)  None,  except  to  illustrate  solid  bodies.  (6)  Extensively  and  suc- 
cessfully. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.   337 

(a)  To  what  extent  are  models  used  in  geometry  ?  (b)  To  what  extent  and  ivith  tohat  success 
original  exercises  ?  (c)  Do  you  favor  memorizing  verhatim  the  theorems  (not  the  demon,' 
strations)  in  geometry  f     What  reforms  are  needed  in  teaching  the  same  ? — Continued. 

25.  (a)  From  want  of  fnnds  the  supply  is  limited  to  such  rude  models  aa 
teacher  and  student  can  make,  (b)  Original  exercises  in  connection  with 
every  study  are  used  freely  and  with  good  results,  (c)  More  original  exer- 
cises. 

26.  (e)  No.    Variation  of  letters,  etc.,  to  represent  angles. 

27.  (a)  I  have  relied  upon  models  to  a  great  extent.  I  require  all  studying 
solid  geometry  to  construct  the  five  regular  polyhedrons  with  pasteboard,  giv- 
ing reason  why  only  five  can  be  formed.  (&)  All  classes  work  at  original 
propositions.    The  results  have  shown  the  practice  to  be  very  important. 

28.  (a)  No  models  used  in  plane  geometry.  The  sphere,  the  cone,  and  a  few 
others  are  used  in  solid  geometry,  (b)  One-half  of  all  the  time  for  geometry 
is  devoted  to  original  exercises.  Success  very  satisfactory,  (c)  Some.  More 
original  work.  Demonstrations  varying  from  those  of  the  author  should  be 
encouraged. 

29.  (a)  Not  to  a  great  extent,  (h)  They  are  freely  used  with  the  best  results, 
(c)  Yes,  substantially;  the  gravest  error  is  the  memorizing  of  demonstrations — 
an  evil  that  seems  unavoidable,  if  text-books  are  employed.  The  ideal  method 
is  oral  instruction,  in  which  the  mental  movements  of  the  pupils  are  under  the 
eye  of  the  instructor.  It  is  a  pity  that  a  subject  that  has  such  possibilities  for 
pupils  should  be  so  taught  as  to  become  a  mere  "  memory  grind." 

30.  (a)  To  no  great  extent,  but  figures  extensively.  (&)  No  great  extent. 
Human  nature  is  not  original.  Originality  is  the  exception.  The  average 
student  who  spends  his  time  on  original  exercises  will  fail  of  that  discipline  in 
method  which  he  needs,  (c)  Yes,  the  student  will  become  benefited  in  learning 
of  a  formula  of  words  expressing  truths.  Stick  to  the  Euclidian  method ;  there 
is  no  "royal  road"  to  geometry. 

31.  (a)  Such  as  we  are  able  to  improvise. 

32.  (a)  They  are  not  used,  (b)  We  intersperse  them  throughout  the  entire 
course  of  geometry,  (c)  "While  they  should  be  memorized,  the  student  should 
learn  to  state  them  also  in  good  language  of  his  own. 

33.  (a)  Practically  to  no  extent.  (&)  Great  success  when  used,  (c)  More, 
much  more  original  work  and  simplification  of  demonstrations. 

34.  (a)  Very  little.  (6)  One-third  to  one-half  of  work  assigned.  Great  suc- 
cess with  the  better  students,     (c)  No. 

35.  (a)  Used  in  the  study  of  the  geometry  of  space,  of  surifaces  of  the  second 
order.  (6)  Greometry  is  crowded  into  short  time  for  necessary  reasons ;  some 
original  work  done,  less  than  would  be  useful,  (c)  Yes.  Greater  familiarity 
with  definitions  and  axioms.  The  constructive  method  of  carrying  on  demon- 
strations {i.e.,  omit  drawing  figure  in  full,  beforehand). 

36.  (a)  We  approve  of  their  extensive  use.  (&)  Throughout  the  course ;  sue. 
cess  indifferent,  (c)  Yes.  More  attention  to  the  form  of  demonstration  and  ac- 
curacy of  statement. 

37.  (a)  Not  used  to  any  extent.  (6)  Original  exercies  are  extensively  used, 
and  a  greater  interest  in  the  study,  (e)  More  attention  paid  to  original  exer- 
cises well  graded. 

38.  {a)  Models  are  used,  (b)  All  the  examples  in  Wentworth's  Geometry  are 
Bolved,  together  with  selections  outside.  We  are  more  successful  each 
succeeding  year,  (c)  No.  In  general,  less  text-book  routine  and  more  prob- 
lems, not  so  difficult,  but  well  graded. 

39.  (a)  Very  little.  (&)  Much  and  with  great  success,  (c)  Yes.  More  demon- 
strations should  be  written  out,  both  in  the  elements  and  among  original  exer- 
cises. 

881— No.  3 ^22 


338  TEACHING  AND   HISTORY   OF  MATHEMATICS. 

(a)  To  what  extent  are  models  used  in  geometry  ?  (6)  To  what  extent  and  ivith  what  success 
original  exercises  ?  (c)  Do  you  favor  memorising  verhatim  the  theorems  {not  the  demon- 
strations) in  geometry  ?     JVhat  reforms  are  needed  in  teaching  the  same? — Continued. 

40.  (a)  Not  very  much.  I  prefer  tliat  students  should  learn  as  soon  a,s  possi- 
ble to  form  mental  pictures  of  the  figures  and  reproduce  them  on  the 
board.  (&)  Frequent  original  problems  are  given  and  are  very  valuable,  (c) 
Original  problems  and  propositions  should  be  given  in  connection  with  the  les- 
sons from  the  beginning. 

41.  (a)  They  are  used  in  teaching  the  higher  surfaces,  especially  the  warped 
surfaces  in  descriptive  geometry.  (6)  A  few  original  exercises  are  given  with 
the  text-book  work,  and  with  marked  success,  (c)  No.  More  reliance  upon 
the  imagination  for  the  figures  and  less  upon  the  blackboard. 

42.  (a)  Limited.  (&)  Very  largely  and  with  great  success,  (c)  Not  neces- 
sarily. 

43.  (a)  A  limited  extent.  (&)  Original  demonstrations  are  required  on  one 
day  of  each  week  of  second  term,     (c)  Yes, 

44.  (J))  From  first  to  last  with  good  success,  (c)  I  do  not.  More  original 
work. 

45.  {a)  They  are  all  represented  by  the  blackboard.  (&)  As  much  as  possi- 
ble ;  usually  daily,  (c)  I  do.  Less  demonstrations  in  the  book ;  more  propo- 
sitions for  the  student. 

46.  (a)  Geometry  is  with  us  a  preparatory  study.  (6)  Constantly,  with  suc- 
cess, (c)  No.  The  geometrical  teaching  in  our  public  schools  seems  to  be  ex- 
cellent. 

■  47.  {a)  To  a  large  extent.  (&)  With  good  success  when  in  printed  form; 
otherwise  not  so.  (c)  Unless  the  student  can  hold  himself  to  an  equally  clear 
form.    A  union  of  the  old  rigor  with  modern  improvements. 

48.  (a)  The  usual  bloclis,  etc.  (&)  About  one  exercise  out  of  every  ten  with 
fair  success,     (c)  If  the,  text  is  given  in  defmite  form  and  is  well  worded,  yes. 

49.  (a)  None,  excepting  the  elementary  models,     (c)  In  part. 

50.  (a)  As  far  as  needed  in  all  cases.  (&)  Not  much  success  as  yet,  but  hoije- 
ful.     (c)  I  do.    More  use  of  exercises  and  original  work. 

51.  (a)  We  have  about  one  hundred  dollars'  worth  of  models  for  pure  mathe- 
matics,     (c)  No. 

52.  {a)  For  solid.  (&)  From  the  first  and  with  gratifying  success,  (c)  Yes. 
Practical  application  of  principles  in  concrete  problems. 

53.  (a)  Only  ordinary  models,  or  those  commonly  used.  (6)  Occasionally  with 
good  success,     (c)  Yes.    Less  speed  and  more  thoroughness. 

54.  (fl.)  To  a  limited  extent.  (&)  Made  prominent  and  with  good  effect,  (c) 
To  discourage  mere  effort  to  demonstrate  by  memory,  rather  than  by  intuition 
and  train  of  reasoning. 

55.  (a)  As  far  as  possible,  especially  in  solid  geometry.  (&)  A  great  many 
original  exercises.  They  are  the  best  measures  of  the  student's  ability,  (c)  No. 
Less  memorizing  of  demonstrations  and  more  original  work. 

56.  (J))  The  representative  theorems  are  all  demonstrated  by  original  work 
as  far  as  possible,  (c)  Yea.  Any  plan  that  will  prevent  students  from  memoriz- 
ing the  demonstrations. 

57.  (ft)  Not  at  all.  (I)  Have  not  tried  this  plan  yet.  (c)  No.  Not  prepared 
to  suggest. 

58.  (a)  Not  much,  and  mostly  in  spherical  geometry.  (6)  Have  had  some 
original  work  with  profit,  (c)  Yes.  More  familiarity  with  .relations  of  parts 
to  each  other,  and  less  dependence  on  the  wording  of  the  demonstrations  as 
given  in  the  book. 

59.  (a)  Wc  usually  use  diagrams.  (&)  To  considerable  extent  and  with  emi- 
nent success,  (c)  Yes.  All  hail!  to  the  man  who  will  devise  means  to  prevent 
the  pupil  from  committing  to  memory  the  demonstrations , 


MATHEMATICAL    TEACHING   AT    THE    PRESENT    TIME.        339 

(a)  To  what  extent  are  models  iised  in  geometry  ?  (6)  To  tvTiat  extent  and  with  what  success 
original  exercises?  (c)  Do  you  favor  memorizing  verhaiim  the  theorems  (not  the  demon-^ 
strations)  in  geometry  f     What  reforms  are  needed  in  teaching  the  same  ? — Continuecl. 

60.  (a)  To  a  moderate  extent.  (&)  Largely,  and  with  great  success,  (c)  Yes. 
Guarding  against  use  of  memory  too  much  by  stndeuts  in  demonstrations  of 
propositions. 

61.  (a)  In  lecturing  only,  (c)  Yes.  Thorough  understanding  of  rel'Atively 
important  principles. 

62.  (a)  To  a  very  limited  extent.  (&)  We  use  a  great  many  original  ^jxercises 
with  much  success,  (c)  Yes.  Too  many  allow  students  to  memorize  tl  le  demon- 
strations and  thus  miss  the  great  advantage  in  geometry,  a  develoj^n^ient  of  the 
reasoning  faculties. 

;  G3.  (a)  So  far  as  to  illustrate  triangles,  parallelograms,  circle^y,  pyramids, 
prisms,  cones,  cylinders,  and  spheres.  (&)  Limited,  (c)  Yes.  Dejmonatrations 
ought  not  to  be  memorized.  Pupils  ought  to  be  shown  that  the.  truth  of  each 
proposition  is  established  by  a  course  of  logical  reasoning. 

64.  (a)  For  illustrating  solid  geometry,  mensuration,  conic  se&tions.     (c)  Yes. 

65.  (a)  Not  at  all.  This  is  due  to  the  school  not  being  provided  with  models 
and  not  to  the  teacher's  preference.  (5)  They  are  used  only  occasionally,  but 
with  considerable  success  when  used,     (o)  Yes. 

66.  (ffl)  Whenever  necessary,     (c)  Yes. 

68.  (a)  Not  at  all  in  geometry,  to  slight  extent  in  descri-ptive  geometry.  (6) 
Cadets  are  frequently  required  to  submit  exercises,   (c)    No. 

69.  (a)  Largely.  (6)  Few,  but  satisfactory,  (c)  No.  Latitude— bo  long  as 
object  is  clearly  stated,  and  demonstration  is  concise  and  complete. 

70.  (a)  Very  limited.  (6)  To  a  limited  extent,  but  with  good  success,  (c) 
Yes.    More  extended  use  of  models. 

71.  (a)  The  spherical  blackboard  and  models  are  used  considerably.  (&)  They 
are  being  introduced  with  good  success,  (c)  No ;  except  for  those  students  who 
must  in  order  to  understand  them.  Students  should  learn  to  depend  less  on 
the  printed  denonstrations. 

72.  (a)  Definitions  are  taught  by  means  of  models,  (b)  The  extent  varies 
with  different  classes.  The  success  is  good  with  about  one-third  of  the  class, 
(c)  No. 

73.  (a)  Little.     (&)  To  some  trifling  extent,  always  with  profit,    (o)  Yes. 

74.  (a)  Where  models  seem  to  make  principles  clearer,  or  their  application 
practical,  they  are  used  in  teaching  solid  geometry  and  spherical  trigonometry, 
(c)  Yes. 

75.  (6)  Nearly  one-half  the  time  is  spent  upon  original  work  and  with  marked 
success,    (c)  Not  absolutely. 

76.  (a)  In  metrical  geometry  models  are  used  altogether  for  illustration. 
(&)  Our  time  being  limited,  we  spend  little  on  original  exercises,  but  with 
fine  success,  (c)  Yes.  The  student  should  be  required  to  carefully  write  each 
demonstration  upon  the  board. 

77.  (a)  In  solid  geometry  all  the  principal  figures  are  thus  illustrated.  (6) 
To  a  considerable  extent  in  plane  geometry  and  with  excellent  success,  (c) 
Yes.    More  original  work  and  less  memorizing  of  demonstrations. 

78.  (a)  Very  little.  (&)  To  a  very  considerable  extent  and  with  marked  suc- 
cess,   (c.)  No.    More  attention  should  be  given  to  original  work. 

79.  (a)  To  no  great  extent.  (&)  Original  exercises  are  given  as  optional  work 
and  a  few  students  are  very  successful  in  them,  (c)  Indifferent,  provided  they 
are  given  clearly  and  concisely. 

89.  (a)  I  use  them  very  frequently.  (&)  Original  exercises  form  a  part  of 
nearly  every  lesson.  With  a  few  exceptions  the  results  are  excellent,  or  at 
least  satisfactory,    (c)  Yes.    More  original  work. 


340  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

(a)  To  liiMt  extent  are  models  used  in  geometry  ?  (i)  To  what  extent  and  ivitTi  what  success 
original  exercises  ?  (c)  Do  you  favor  memorising  vertatim  the  theorems  {not  the  demon- 
strations) in  geometry  ?     What  reforms  are  needed  in  teaching  the  same  f — Continued. 

82.  (a)  Somewhat  in  solid  geometry,  (h)  To  a  very  large  extent  in  daily 
■work  and  with  very  satisfactory  results,  (c)  I  hoi  I  students  responsible  for  a 
knowledge  of  the  theorem, but  not  verbatim. 

83.  (i)  Original  exercises  are  used  and  with  good  success,     (c)  Yes. 

84.  (a)  Not  at  all.  (&)  To  a  considerable  extent  and  with  much  success,  (o) 
No.    More  attention  to  logical  form  and  precision  of  statement. 

i.'SS.  (a)  In  course  on  "  form."  (b)  As  far  as  the  time  allotted  will  allow,  and 
with  great  success,  (c)  Yes.  A  greater  use  of  objects.  A  leaving  of  parts  of 
the  demonstrations  to  be  filled  in,  thus  training  for  origin  al  work. 

86.  (c)  No.    The  adoption  of  the  heuristic  method. 

87.  {a)  Very  little.  (6)  Original  exercises  comprise  a  very  large  part  of  the 
•work,  «ay  one-half,  in  geometry,  (c)  Yes ;  those  to  be  frequently  referred  to  in 
Bubseqtxent  work ;  others,  no. 

88.  (6)  Much  used  in  geometry,  and  very  successfully,    (c)  Yea. 

89.  (rt)  None.  (J)  Numerous  practical  problems  with,  I  think,  good  success, 
(c)  No. 

90.  (a)  To  a  small  extent.  (J)  A  very  large  extent  and  excellent  results,  (c) 
No.  Cultivation  of  more  originality  by  means  of  graded  exercises. 

91.  (a)  They  are  not  much  used.  (6)  They  are  used  whenever  there  is  an 
opening;  success  is  good,    (c)  No. 

92.  (o)  They  are  used  to  some  extent  in  solid  and  general  geometry.  (6)  To 
considerable  extent,  with  very  satisfactory  results,  (c)  As  a  rule,  yes ;  for  the 
reason  that  they  are  usually  stated  much  more  concisely  than  the  student  would 
state  them.    More  original  demonstrations. 

93.  (a)  Full  sets  of  Schroder's  (Darmstadt)  models.  In  solid  geometry  stu- 
dents make  models  from  pasteboard,  (b)  Such  exercises  in  connection  with 
nearly  every  lesson,  and  with  gratifying  success,  (c)  Yes ;  number  of  propo- 
sition and  book  should  not  be  memorized.  More  problems  and  practical  appli- 
cations :  more  theorems  for  original  demonstration  by  pupils. 

94.  (a)  Very  sparingly ;  find  them  hurtful  rather  than  helpful.  "Normal 
school"  methods  are  a  failure  in  geometry.  Have  tried  both  and  seen  both 
tried.  (J)  In  connection  with  nearly  every  theorem  and  every  lesson.  Success 
good,  (c)  Yes — no,  depends  on  the  student  and  the  sort  of  drill  he  needs.  A 
more  rigorous  insistence  on  founding  everthing  on  the  axioms. 

95.  (&)  Many  new  exercises  with  great  success,    (c)  Introduce  more  exercises  , 
and  require  variation  in  figures. 

96.  (a)  None.  (6)  Very  considerable  extent  and  good  success,  (c)  Not  ver- 
batim, but  clearly  and  fully  in  substance. 

97.  (a)  Hitherto  but  little ;  henceforth  very  great  (if  the  appropriation  asked 
of  the  State  be  granted).  (&)  If  unassisted,  or  only  slightly  assisted,  demonstra- 
tion and  solution  be  meant,  great  and  good,  (o)  No  !  liupture  ivith  the  tradi- 
tional Euclidian  methods,  alignment  with  the  march  of  modern  thought. 

98.  (a)  To  a  small  extent  in  descriptive  geometry  (warped  surfaces,  etc.). 
•    (6)  To  a  great  extent  and  as  much  as  possible,  and  with  marked  success,     (c) 

No,  only  to  a  slight  extent  for  beginners.    More  original  exercises,  and  more 
modern  geometry  of  position. 

99.  (a)  Average.     (&)  Tested  daily,     (c)  Yes. 

100.  (a)  We  use  models  of  the  usual  geometrical  forms  for  illustration,  (b) 
Frequent  exercises  in  geometry ;  success  only  moderate,     (c)  No. 

101.  (a)  In  solid  geometry,  (b)  One-third.  Good  results  with  fair  success. 
(c)  Yes.    Larger  per  cent,  of  original  work  required. 

102.  (a)  None,  (b)  Extensively  used,  and  results  very  gratifying,  (c)  Yes. 
More  original  exercises  and  a  more  rigid  reference  to  first  principles. 


M 


MATHEMATICAL    TEACHING   AT    THE   PRESENT   TIME.         341 

(a)  To  lohat  extent  are  models  used  in  geometry?  (h)  To  wliat  extent  and  with  tvliat  success 
original  exercises  ?  (c)  Do  you  favor  memorizing  veriatim  the  theorems  (not  the  demon- 
strations) in  geometry  ?    What  reforms  are  needed  in  teaching  the  same? — Continued. 

103,  (a)  None.  (&)  In  geometry,  with  fair  success ;  in  practical  surveying, 
leveling,  etc.    (c)  Yes.    As  a  rule,  not  allowing  any  lettering  on  board,  etc. 

104.  (a)  None.     (&)  None,     (c)  No. 

106.  (a)  To  a  very  limited  extent.  (&)  To  the  extent  which  the  time  will 
permit,  and  with  increasing  degree  of  success,  (c)  I  do  not.  I  think  reform 
needed  in  regard  to  grasping  the  truth,  and  giving  it  in  good  language,  of  the 
pupil's  selection. 

107.  (a)  Very  little,    (b)  Slight  extent,     (c)  No.     More  original  work. 

108.  (a)  Small,     (o)  Yes. 

109.  (a)  Very  little.  (&)  Considerable,  with  fair  success,  (c)  No.  Incorpo- 
ration of  some  treatment  of  modern  geometry. 

110.  (a)  To  a  considerable  extent  in  teaching  solid  geometry  and  spherical 
geometry.  (&)  Original  exercises  constitute  half  of  the  work,  and  with  satis- 
factory success,  (c)  Yes.  Eequire  more  solutions  of  practical  problems ;  this 
tests  the  ability  of  the  student  and  teaches  him  to  walk  alone. 

111.  (a)  None.     (&)  We  use  few.     (c)  It  is  not  material  with  us. 

112.  (a)  Only  to  a  limited  extent  in  illustrating  some  of  the  properties  of 
planes  and  solids.  (6)  To  a  great  extent,  and  with  satisfactory  results,  (c) 
I  do.  It  is  necessary  that  the  student  should  know  what  he  is  required  to 
demonstrate.    Theory  and  practice  should  go  hand  in  hand. 

113.  (a)  They  are  used  for  every  demonstration  in  solid  geometry,     (e)  No. 

114.  (a)  Very  little.  (&)  It  has  not  seemed  profitable  to  spend  much  time  on 
original  work  in  geometry,  which  is  a  study  of  the  Freshman  year,     (c)  No. 

115.  (i)  A  great  number  of  original  exercises  are  given  with  complete  suc- 
cess, (c)  I  do  not  favor  memorizing  anything  except  such  principles  as  are 
needed  in  after  work.  For  purposes  of  illustration  we  have  a  full  set  of  models 
of  solid  and  descriptive  geometry,  (i)  Original  exercises  are  given  as  regular 
and  extra  work  to  all  classes,  (c)  Yes.  The  elementary  principles  of  logic 
should  be  explained  in  connection  with  elementary  geometry.     (Tutor  Fiske.) 

IIG.  (a)  None.  (&)  Increasing  number  from  year  to  year.  Successful  with 
first  third  of  the  class. 

117.  (a)  Students  make  models  in  solid  geometry,  (h)  With  excellent  results 
and  to  a  large  extent  in  geometry  and  trigonometry,  (c)  Yes.  Throw  students 
more  on  their  own  resources. 

118.  (a)  Merely  in  explaining  and  illustrating,  (b)  They  are  required  more 
or  less  throughout  the  course,  especially  in  geometry,     (c)  Yes. 

119.  (a)  We  use  the  globe  and  the  usual  geometrical  solids.  (&)  Original 
exercises  are  frequently  given  ;  success  is  very  fair,     (c)  Yes. 

120.  (a)  None  are  used  in  plane  geometry.  We  have  twenty-six  fine  models 
of  warped  and  single-curved  surfaces  for  use  in  descriptive  geometry.  (6)  Orig- 
inal exercises  are  given  out  at  each  recitation,  with  great  success  as  regards 
the  development  of  mathematical  knowledge,  (c)  No.  With  each  lesson  the 
student  should  have  several  original  exercises  involving  the  principles,  to  solve 
or  demonstrate. 

121.  (a)  Always  in  the  teaching  of  geometry  of  space,  when  I  find  it  helpful. 
(6)  Continually  given  as  voluniai'y  text-vfoi^,  excusing  the  student  from  formal 
examination  in  proportion  to  her  success  in  it.  (c)  No.  The  abolition  of  in- 
direct proof,  and  the  use  of  symbolic  notation,  with  special  attention  to  form. 

122.  (a)  In  solid  geometry,  (b)  To  a  limited  extent,  (c)  Yes.  More  fre- 
quent direct  application  to  problems  in  which  dimensions  are  to  be  found. 

123.  (o)  To  a  moderate  extent.  (&)  To  a  considerable  extent  and  with  as 
good  success  as  can  be  expected,  (c)  Yes.  More  frequeat  tests  on  original 
theorems  and  problems. 


342  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

(c)  To  ivhal  extent  are  models  used  in  geometry  f  (&)  To  wliat  extent  and  with  what  success 
original  exercises?  (c)  Do  you  favor  memorizing  veriatim  the  theorems  {not  the  demon- 
strations) in  geometry  ?     What  reforms  are  needed  in  teaching  the  same  ? — Contiuued. 

124.  (a)  The  students  make  models  of  tlie  regular  polyliedrous.  (h)  Large 
numbers  given  ia  each  class ;  this  is  one  of  our  chief  methods  of  drill  and 
training  ;  it  is  the  only  tvay  in  which  fundamental  princij)les  can  he  so  thoroughly 
ingrained  in  the  mental  maTce-up  of  a  student  that  he  is  no  longer  conscious  of  an 
effort  of  memory  in  Ms  Tcnowledge.  (c)  Yes,  to  encourage  exact  expressions  and 
as  a  tribute  to  order  which  is  the  soul  of  geometrical  reasoning. 

125.  (a)  Very  little  at  present;  hope  to  use  them  extensively.  (&)  To  as 
large  an  extent  as  possible.  With  good  success  from  the  majority  of  students, 
(c)  I  do  not.  (1)  Better  trained  teachers;  (2)  more  thinking  and  less  memo- 
rizing; (3)  use  of  thoroughly  good  text-books,  like  Byerly's  Chauvenet;  (4) 
emphasis  of  logic  side  ;  (5)  generalization  and  summing  up  of  truths  proved,  etc. 

126.  (a)  No  geometry  taught  excepting  descriptive  geometry.  Students  con- 
struct their  own  models.  (&)  Original  exercises  in  almost  daily  use.  (c)  Yes. 
Demonstration  of  theorems  without  letters  or  figures. 

127.  (a)  We  have  a  set  of  "geometrical  solids,"  which  we  use  on  occasion, 
(fe)  Such  exercises  are  often  required,  and  they  are  valuable — increasiug  the 
interest  and  testing  the  student's  knowledge. 

128.  (a)  Not  largely,  but  so  far  as  the  students  seem  to  need  them.  (5)  Very 
largely  and  with  good  success,  (c)  A  clean-cut,  accurate  statement,  whether 
verbatim  or  not.  Teacher  to  make  sure  of  actual  mastery  of  principles — no 
memory  work;  much  use  of  original  exercises. 

129.  (a)  So  far  as  is  necessary  for  the  pupil  to  get  a  clear  conception  of  the 
geometrical  concept.  (&)  In  the  preparatory  course  for  admission  to  Fresh- 
man class,  limited.    In  Freshman  used  to  large  extent,     (o)  Yes. 

130.  (a)  Models  of  solids  are  used  in  solid  geometry.  (&)  A  good  deal  of  use 
is  made  of  them.  Success  good  with,  best  students,  (c)  No.  More  time  should 
be  given  to  leading  the  student  to  discover  theorems  for  himself. 

131.  (a)  Not  very  much.  (6)  As  much  as  possible  and  with  gratifying  results, 
(c)  No.    More  original  exercises. 

132.  {a)  Geometry  is  finished  before  entrance,  except  descriptive  geometry,  in 
which  we  use  no  models. 

133.  ia)  To  a  limited  extent  with  sphere  and  regular  polyhedrons.  (&)  The 
original  work  in  Wentworth's  Geometry,  with  fair  success,     (c)  Yes. 

134.  (a)  In  solid  geometry,  (b)  Exercises  and  constructions  on  each  book, 
and  with  good  success,  (c)  No.  Something  like  Wentworth's  system  of  dem- 
onstrating propositions. 

135.  (a)  None  except  sphere  and  cone.  (&)  At  least  one  original  exercise  is 
given  as  a  part  of  each  lesson.  Great  success,  (c)  No.  Students  should  be 
taught  to  master  new  processes  or  methods  of  proof  rather  than  individual 
theorems ;  so  come  to  look  on  theorems  and  proofs  as  illustrations  of  processes, 
or  methods  of  investigation. 

136.  (a)  Not  at  all.  (&)  Very  little,  in  obligatory  mathematics ;  and  "in  reg- 
ular course  there  is  hardly  any  pure  geometry;  but  when  there  is  any,  such 
exercises  are  helpful,  (c)  Only  in  elementary  work,  and  even  in  that  the  at- 
tendant dangers  are  great. 

137.  (a). To  a  slight  extent  only.  (&)  Limited  extent,  but  with  good  success, 
(c)  Yes. 

138.  (a)  To  a  limited  extent.  (&)  They  are  much  used  and  with  good  results, 
(c)  No. 

139.  (a)  Very  sliglit.  (&)  About  fifty  original  exercises  are  given  and  are 
well  done,     (e)  Yes. 

140.  (a)  Slightly.     (&)  Considerably,  with  success,     (c)  No. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.    343 

(a)  To  what  extent  are  models  used  in  cjeometrg  f  (b)  To  what  extent  and  with  what  success 
original  exercises  ?  (c)  Do  you  favor  memorizing  verhaiim  the  theoreins  (not  the  demon- 
strations) in  geometry  ?     What  reforms  are  needed  in  teaching  the  same f— Continued. 

141.  (a)  Small.  (&)  Consideralile  extent  and  •with  commendable  success, 
(c)  Yes. 

142.  (&)  With  good  success,  as  a  rule,     (c)  Yes. 

143.  (a)  Our  models  are  sucli  as  we  make  ourselves.  Wo  illustrate,  so  far  as 
possible,  in  solid  and  descriptive  geometry,  (i)  All  that  I  can  bave  time  for 
and  can  get  tba  students  to  solve ;  great  success  with  the  few,  little  with  the 
mass,  (c)  No ;  but  I  require  a  clear  statement  in  somebody's  Avords.  More  origi- 
nal work  should  be  given  ;  the  student  should  be  taught  to  depend  upon  him- 
self more,  and  less  upon  book  or  teacher,  to  tUnlc,  to  originate,  not  memorize, 
not  absorb. 

J44.  (a)  Not  at  all,  except  for  young  pupils.  (6)  Daily  use  and  -with  good 
success,  (c)  Yes.  Young  pupils  should  be  drilled  in  practical  exercises,  with 
use  of  instruments. 

145.  {a)  Have  been  used  but  little.  (&)  Constantly  and  successfully,  (c)  No. 
An  earlier  start,  with  main  attention,  at  first,  to  training  observation.  Greater 
freedom  fromformaliam. 

146.  {a)  Whenever  possible.     (S)  Limited,     (c)  Yes. 

147.  (a)  I  make  considerable  use  of  models,  especially  in  solid  geometry.  (5) 
Original  exercises  are  required  at  a  few  places  (two  or  three  lessons).  Good 
success,  (c)  Yes  ;  yet  I  do  not  insist  on  keeping  every  word,  provided  the  sense 
is  kept. 

148.  (a)  We  do  not  use  models.  (&)  We  lay  great  stress  on  original  exercises. 
When  properly  selected  they  are  of  the  utmost  service,  (c)  No.  A  clearer  com- 
prehension of  the  definitions ;  a  more  frequent  enumeration  of  facts  already 
proved;  a  more  explicit  enumeration  of  facts  to  be  established  in  demonstrating 
any  particular  theorem. 

149.  (a)  In  a  very  limited  degree.  (&)  For  the  past  few  years  I  have  used 
them  freely  -with  gratifying  success,     (c)  No. 

150.  («)  To  a  limited  extent,     (c)  No. 

151.  (&)  The  study  of  geometry  would  fall  far  short  of  its  object  if  original 
work  were  not  required.  I  devote  one  recitation  hour  each  week  to  it,  and  I 
am  pleased  with  the  results.  I  judge  of  the  mental  development  by  the  orig- 
inal work  done  by  pupils,  (c)  Yes.  The  facts  of  geometry  must  come  first, 
concrete  object  lessons ;  can't  teasou  about  that  concerning  which  we  know 
little  or  nothing. 

152.  {a)  But  little,  except  in  descriptive  geometry.  (Z>)  Much  time  is  given 
to  solution  of  problems,  both  from  text-book  and  from  other  sources,     (c)  Yes. 

153.  (a)  Largely,  especially  in  solid  geometry.  (&)  Continually  and  copiously 
and  with  great  success,  (c)  No.  The  rejection  ofthetvords  "direction"  and 
"  distance  "  from  elementary  geometry. 

154.  (a)  Largely,  especially  in  conic  sections  and  descriptive  geometry.  (6) 
Weekly  exercises  and  with  very  satisfactory  results,     (c)  No. 

155.  (a)  To  a  limited  extent.  I  expect  to  use  models  to  a  greater  extent  in 
the  future.  (&)  Original  exercises  are  greatly  used.  I  value  them  very  highly, 
and  I  am  much  pleased  with  the  results  I  have  obtained  by  using  them  in  all 
my  classes,     (c)  I  do  not. 

156.  (a)  Noneused.  (&)  Special  prominence  is  given  to  the  use  of  original  ex- 
ercises, with  encouraging  success,     (c)  Yes.    Originality  should  be  encouraged. 

1.57.  («)  They  are  used  to  a  very  limited  extent,  simply  because  the  college 
is  not  supplied  with  them.  (&)  Frequent  exercises  are  given  with  quite'  good 
success,  (c)  I  do  not ;  I  think  that  a  student  should  be  required  always  to  ex- 
press his  thoughts  in  his  own  language,  if  for  no  other  reason  than  to  acquire 
facility  in  expression. 


.344  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

(a)  To  ivhat  extent  are  models  used  in  geometry?  (&)  To  ivTiat  extent  and  with  ivJiat  success 
original  exercises  ?  {c)  Do  you  favor  memorizing  verbatim  the  theorems  (not  the  demon- 
strations) in  geometry?     What  reforms  are  needed  in  teaching  the  same? — Continued. 

158.  (a)  To  only  a  small  extent.  (&)  Largely  given,  and,  I  think,  with  great 
success  in  promoting  intellectual  pluck  and  thoroughness  of  attainment,  (c) 
Yes ;  and  also  to  learn  to  state  them  in  one's  own  words.  Subject  too  large  for 
space.  I  will  say,  however,  that  the  schools  should  give  more  exercises  for 
solution,  and  train  the  boys  from  the  beginning  in  original  solution. 

159.  (a)  Models  are  used  iu  conic  sections  and  for  surfaces  of  revolution  in 
analytic  geometry  of  three  dimensions.  (6)  Special  attention  given  to  original 
exercises.  A  taste  for  such  work  is  easily  developed  in  every  lover  of  mathe- 
matics, (c)  No;  would  prefer  that  the  student  thoroughly  understand  the 
truth  to  be  demonstrated  and  express  same  in  his  own  language. 

160.  (a)  Very  extensively,  especially  in  solid  geometry,  (b)  Largely,  and 
with  decided  success,     (c)  Yes.    I  would  have  it  made  more  practical. 

161.  (a)  None.  (&)  Original  exercises  are  frequent  and  attended  with  encour- 
aging success,  (c)  I  do.  Demonstrations  should  be  less  verbose,  and  expressed 
to  a  greater  extent  by  algebraic  symbols. 

162.  (a)  Considerably.  (&)  With  almost  every  lesson,  and  with  much  success, 
(c)  To  some  extent. 

163.  (a)  Used  to  illustrate  definitions.  (Z>)  We  devote  some  time  to  them  now ; 
shall  devote  more  ;  excellent  success,  (c)  I  do.  Greater  care  that  the  pupil 
understand  the  reasons  for  every  statement. 

164.  (a)  Models  used  but  little  in  first  presenting  the  subject,  (b)  Original 
exercises  given  throughout  the  course,  with  good  success,  (c)  No.  Geometry 
should  be  taught  as  algebra  and  arithmetic  by  original  work. 

165.  (a)  Limited,    {h)  Largely,  with  great  success,     (c)  Yes. 

166.  (a)  We  have  none.  (&)  On  each  recitation,  when  there  is  time,  (c)  Yes. 
Less  text-book  and  more  originality. 

167.  (a)  Very  little.  The  attempt  is  made  to  lead  the  student  to  form  his 
magnitudes  in  space,  and  without  even  a  drawing,  if  possible.  (&)  To  a  very 
considerable  extent,  especially  in  test-work;  and  with  excellent  success  in 
about  one-half  the  cases,  (c)  No;  but  I  would  insist  on  concise  and  accurate 
statement.  (1)  Less  bondage  to  text-book.  (2)  Encouraging  original  demon- 
strations.    (3)  Clearer  distinction  between  leading  steps  of  proof  and  details. 

168.  (a)  Only  slightly,  because  we  are  not  able  to  afford  them.  (&)  They  are 
used  as  much  as  time  will  permit,  and  with  good  success,  (c)  No.  Less  time 
given  to  theorem-demonstration  and  more  to  original  exercises,  with  tha  proper 
change  in  text  books. 

Is  elementary  geometry  ^preceded  or  accompanied  hy  drawing  ? 

"Preceded" :  21,  22,  29,  49,  51,  53,  55,  58,  62,  73,  76,  82,  90,  92,  98,  102,  103,  110, 
112,  114,  139,  147. 

"  Accompanied : "  8,  10,  12,  13,  14,  20,  24,  25,  27,  28,  30,  33,  39,  41,  42,  43,  44,  48,  54, 
56,  57,  59,  60,  61,  66,  70,  71,  84,  94,  96,  97,  99,  101,  105,  106,  122,  127,  134,  135,  137,  138, 
140,  142,  150,  153,  154,  457,  159,  164,  165,  166. 

"  Both  preceded  and  accompanied":  4,  9,  11,  69,  85,  87,  88, 107,  111,  119,  124,  146, 
160,  161. 

"Neither  preceded  nor  accompanied":  1  (except  engineering  students),  15,  16, 
17,  19,  31,  32,  34,  35,  37, 38,  45,  63,  77  (except  in  industrial  course),  81,  91,  93,  100,  108, 
109, 115,  116,  117,  120,  125,  128,  129  (except  in  scientific  course),  130, 131,  149,  150,  155, 
156,  167,  168. 

"Yes;  either  preceded  or  accompanied" :  23,  26,  123,  145, 163. 


MATHEMATICAL   TEACHING   AT   THE   PEESENT   TIME.        345 

Statetime  of  your  s;pecial  preparation  for  teaching  mathematics,  nuviier  of  hours  you  teach 
per  tveelc,  and  what  other  subjects  you  teach. 

I.  Teach  twenty  liours  per  week ;  teach  no  other  subjects. 

3.  About  ten  years  before  entering  the  University  of  Alabama,  where  I  spent 
five  years ;  physics  and  astronomy. 

4.  I  teach  twenty-five  hours  a  weelc,  and  during  five  months  in  the  year 
give  about  six  hours  a  -week  to  special  work  in  surveying ;  I  teach  no  other 
subjects. 

5.  Six  and  a  half  Tiours  per  day  in  the  entire  school,  with  one-half  hour 
recitation  for  classes  at  diiferent  times. 

6.  From  1880  to  1886 ;  nine  hours ;  geology,  astronomy,  elocution. 

8.  Two  years;  thirty-five  hours  per  week;  book-keeping,  six  hours  per 
■week. 

9.  I  teach  only  mathematics,  and  give  five  lessons  of  one  and  one-fourth  hours 
each  per  week, 

10.  Four  years;  seventeen  hours  per  week;  no  other  subjects  save  military 
Bcience  and  tactics. 

II.  Five  years'  study  at  the  Cincinnati  Observatory  after  graduating  from 
college.  At  present  I  teach  thirty  recitation  hours  (forty-five  minutes  each)  a 
■week.    Astronomy  (popular). 

12.  It  has  been  my  specialty  for  eight  years ;  twenty  hours  per  week ;  none. 

13.  Pare  mathematics  is  taught  twenty  hours  per  week. 

14.  Teach  mathematics  three  and  four  hours  per  week ;  my  chief  subject  is 
chemistry,  -while  mathematics  ia  a  secondary  subject  here. 

15.  I  have  taught  mathematics  since  graduating  from  college  (1870).  I 
teach  from  eight  to  twelve  hours  per  week.    Astronomy  is  also  in  my  charge. 

16.  I  teach  from  twenty  to  thirty  hours  per  week,  and  teach  no  other  sub- 
ject. 

17.  Five  years ;  twenty  hours  per  week ;  none. 

18.  The  classes  of  mathematics,  except  the  first  class,  are  taught  by  the 
professors  of  the  regular  classical  course.  Each  professor  teaches  only  one 
class  of  mathematics. 

19.  Ten  to  fifteen  hours  per  week,  mental  science  and  chemistry,  etc.  (Profes- 
sor Gordon).  Preparation  has  nearly  all  been  made  since  I  began  and  while 
teaching;  ten  hours;  Latin  (Professor  Draper). 

20.  I  teach  mathematics  twenty  hours  per  week.    I  teach  no  other  subjects. 

21.  Mathematics  was  my  specialty  in  college  two  years;  teach  it  fifteen 
hours  per  week.    I  teach  no  other  subjects. 

22.  About  three  hours  per  diem  devoted  to  teaching.  Teach,  besides,  chem- 
istry and  elocution. 

23.  I  teach  twenty  hours  per  week,  and  also  military  science  and  tactics. 
25.  Three  hours  per  day.    I  teach  nothing  but  mathematics. 

27.  I  teach  only  pare  mathematics. 
88.  Thirty;  teach  no  other  subject. 

29.  Fifteen  hours  a  week;  physics  five  hours  a  week  for  six  months;  astron- 
omy five  hours  a  week  for  four  months ;  psychology  five  hours  a  week  for  three 
months. 

30.  Sixteen  to  twenty  hours.  I  teach  philosophy,  astronomy,  logic,  moral 
philosophy. 

32.  I  give  fifteen  to  twenty  hours  of  instruction,  and  teach  several  natural 
sciences,  besides  some  German. 

33.  One  year  and  a  half;  twenty  hours ;  no  other  subjects. 

34.  Several  graduate  courses  at  Princeton  College,  and  private  study,  I 
teach  from  eleven  to  fifteen  hours  per  week.  Am  also  engaged  in  teaching 
astronomy.  " 


346  TEACHING   AND   HISTORY    OF    MATHEMATICS. 

State  time  of  your  special  ])rei)araUon  for  isaching  mathematics,  numier  of  hours  you  teach 
2ier  tveelc,  and  tvhat  other  subjects  you  teach — Continued. 

35.  Have  taught  and  laave  been  a  student  in  the  higher  mathematics  the  past 
twenty  years;  fifteen  hours  per  week;  not  any.  (Professor  Shattuck.)  Alge- 
bra, five  hours  per  week  for  two  terms ;  geometry,  same  ;  natural  philosophy, 
physiology,  botany,  English,  rhetoric,  Latin,  Greek.     (Regent  Peahody.) 

37.  One  year;  twelve  hours  i^er  week,  besides  teaching  some  in  preparatory 
department. 

33.  I  teach  mathematics  and  astronomy  about  twenty  hours  per  week ;  no 
other  subjects, 

39.  Fifteen  hours ;  no  other  subjects. 

40.  Teach  about  twenty  hours  per  week.  I  teach  regularly  no  other  subject, 
and  only  the  mathematics  in  the  college  department. 

41.  I  spend  from  one  to  two  hours  per  lesson  on  mathematical  works,  directly 
or  remotely  connected  with  the  recitation.  I  teach  twenty  hours  j)er  week 
and  only  mathematics. 

43.  Teach  twenty  hours  per  week;  teach  no  other  subject  except  astronomy. 

44.  Eighteen  hours  per  week ;  history  and  vocal  music. 

45.  Everything  pertaining  to  or  suggested  by  the  lesson  is  prepared.  All 
lessons  are  five  hours  per  -week  with  one  exception ;  geometry  has  four. 

46.  I  teach  mathematics  and  astronomy  fifteen  hours  per  week ;  assistant 
teaches  mathematics  fifteen  hours  per  week  also. 

47.  After  graduating  at  Madison,  spent  four  years  in  post-graduate  study  at 
Yale  ;  fifteen  to  twenty  hours ;  astronomy  and  political  economy. 

50.  We  have  four  teachers  of  mathematics,  who  spend  about  fifteen  hours 
fi/  week  in  their  classes. 

51.  Am  at  it  nearly  all  my  time  that  can  be  secured  from  other  work ;  first 
term,  fifteen  hours ;.  second  term,  twenty  hours. 

52.  Thirty  hours  per  week ;  political  economy. 

53.  All  afternoon  for  preparation ;  twenty-five  a  week ;  none. 

55.  Have  taught  it  for  eighteen  years,  six  years  exclusively.  Teach  mathe- 
matics twenty  hours  per  week  ;  teach  nothing  else. 

56.  I  took  the  two  years'  collegiate  course  required  and  took  the  post-gradu- 
ate course,  spending  three  months  on  special  work  in  mathematics ;  fifteen  ' 
hours ;  natural  science. 

57.  Some  class  almost  every  hour  in  the  day,  averaging,  perhaps,  twenty-five 
hours  per  week. 

58.  Scarcely  any  two  terms  the  same. 

59.  About  three-fifths  of  my  time  is  given  to  mathematics  and  about  two- 
fifths  to  Latin  and  Greek. 

60.  Three  years ;  twenty  hours ;  political  science,  astronomy,  and  German. 

61.  Fifteen  hours  per  week ;  mental  philosophy  and  logic. 

62.  I  teach  mathematics  fifteen  hours  per  week  and  have  some  classes  in  Latin. 
My  preparation  is  done  each  night  before  the  work  of  the  following  day. 

63.  Six  to  ten  hours;  eighteen  hours ;  Latin,  physiology,  physical  geography, 
English  literature. 

64.  One  hour  daily ;  twelve  hours  per  week ;  astronomy,  natural  philosophy, 
chemistry,  geology,  mineralogy,  drawing. 

65.  Four  years  at  college,  one  in  private  work,  and  two  at  Johns  Hopkins 
University.    Twenty-four  hours  per  week.    Nothing  else. 

66.  Four  hours  per  week. 

70.  Mathematical  course  at  Yale  College  together  with  four  years  subsequent 
study.    Political  economy  and  English  literature. 

71.  Since  completing  the  course  in  this  college,  three  years  ago,  I  have  spent 
in  i)rivate  study  a  considerable  portion  of  my  time ;  five  to  ten  hours  per 
week ;  rhetoric  and  drawing. 


MATHEMATICAL   TEACHING   AT   THE   PEESENT   TIME.        347 

Staie  time  of  your  special  preparation  for  teaching  mathematics,  number  of  hours  you  teach 
per  iveel;  and  what  other  subjects  you  teach — Continued. 

72.  Eight,  six,  or  four  hours  per  week,  according  to  term,  -whether  it  be  fall, 
winter,  or  spring.    Lecture  on  art. 

73.  The  only  special  preparation  I  employ  is  the  light  reading  of  new  text- 
books which  come  to  hand.  For  over  twenty  years  have  had  no  difficulties  in 
mathematical  instruction.  I  also  teach,  as  occasion  calls,  metaphysics,  morals, 
political  economy,  history,  literature,  etc. 

74.  Teach  on  an  average  twelve  hours  per  week. 

76.  Three  years ;  about  fifteen  hours  per  week ;  physics  and  meteorology. 

77.  I  teach  twenty-five  hours  per  week,  five  hours  of  which  are  devoted  to 
industrial  drawing. 

80.  Three  years  in  which  I  did  the  four  years'  work  in  college  mathematics, 
required  and  elective,  together  with  outside  special  work  in  same  subject.  I 
teach  five  or  six  classes  per  day  in  mathematics.  I  have  only  astronomy,  be- 
sides mathematics. 

82.  Harvard  College,  A,  B.,  with  electives  in  mathematics. 

85.  A  four-years'  course  in  both  State  Normal  school  an*  college ;  no  other 
subject. 

86.  I  teach  mathematics  about  five  to  nine  hours  weekly,  astronomy  three  to 
fifteen  hours  weekly ;  and,  this  term,  am  teaching  algebra.  In  addition,  I  have 
certain  duties  connected  with  the  observatory,  and  a  requirement  of  the  founder 
of  my  professorship,  viz,  I  have  to  contribute  to  the  advancement  of  astronom- 
ical science. 

87.  Two  gentlemen  here  are  occupied  in  teaching  mathematics,  exclusive  of 
analytical  mechanics  and  civil  engineering ;  occupied  in  class-room  eighteen  or 
twenty  hours. 

89.  Five  hours  per  week  and  five  hours  for  assistant ;  mining,  surveying,  me- 
chanics. 

90.  Two  years ;  twenty-two  and  one-half  hours;  no  other  subjects. 

91.  Several  hours  each  day ;  twenty  hours  per  week.  (I  only  teach  the  higher 
branches.)  My  assistant  teaches  all  up  to  and  including  analytic  geometry, 
moral  science,  etc. 

92.  The  mathematics  of  an  ordinary  college  course  and  two  summer  vaca- 
tions' study  with  the  late  Dr.  Edward  Olney,  of  Michigan  University ;  thirteen 
and  one-third  hours  per  week ;  physics  and  astronomy. 

93.  Two  years  (1871-73)  of  post-graduate  work,  and  fifteen  years  since  as 
specialist ;  teach  about  seventeen  hours  per  week ;  part  of  the  work  done  by 
an  assistant ;  astronomy. 

94.  Twelve  and  one-half  hours ;  political  economy. 

95.  I  have  calculus  four  times  per  week,  mechanics  six  times,  and  thermo- 
dynamics (Clausius)  three  times, 

97,  Two  years'  private  study,  three  years'  study  in  Europe ;  fifteen  or  twenty; 
none, 

98,  Five  hours'  class-work  per  week  for  five  years,  full  course, 

99,  Ten  per  week. 

101.  Eighteen  hours  per  week ;  physics. 

103.  Preparation,  four  years'  course  at  the  United  States  Military  Academy ; 
average  time,  ten  hours  per  week  ;  arts  and  science  of  war  and  tactics. 

104.  Forty-nine  years ;  fifteen  hours  per  week ;  no  other  subject. 

105.  The  professor  of  mathematics  has  not  had  special  training,  but  has 
special  aptitude  in  this  direction.    Has  usually  taught  chemistry, 

106.  My  teaching  is  limited  to  mathematics ;  fifteen  hours  per  week, 

107.  Four  years  at  Dartmouth  College,  and  two  jears  at  the  Thayer  School 
of  Civil  Engineering ;  about  ten  hours  per  week ;  mechanics,  astronomy,  me- 
teorology, surveying.  ^ 


348  TEACHING   AND   HISTOEY   OP   MATHEMATICS. 

State  time  of  your  special  preparation  for  ieacMng  mathematics,  numierof  hours  you  teach 
per  tveek,  and  what  other  subjects  you  teach — Continued. 

108.  Twenty  hours  per  week ;  no  other  subject. 

109.  Graduated  Ph.  B.  at  the  University  of  North  Carolina,  and  spent  one 
year  studying  mathematics  at  the  Johns  Hopkins  University ;  eight  hours  per 
week ;  English  four  hours. 

110.  Time  of  teaching  varies  from  eighteen  to  twenty-one  hours  per  week. 

112.  Teaching  hours,  sixteen  per  week ;  I  teach  no  other  subjects. 

113.  Generally  prepare  in  one-half  hoar;  teach  two  hours ;  Latin  and  Greek 
one  hour  each. 

114.  Five  years  ;  fourteen  hours  per  week ;  no  other  subjects. 

115.  Our  professors  teach  but  one  subject. 

116.  My  college  course,  supplemented  by  three  years'  study  in  Germany ;  I 
teach  ten  hours  per  week ;  no  other  subject. 

117.  Have  been  teaching  mathematics  and  science  for  fifteen  years.  At  pres- 
ent most  is  done  here  by  our  instructor. 

119.  From  ten  to  sixteen  hours  per  week  ;  I  teach  nothing  but  mathematics. 

120.  I  dev«fce  eighteen  hours  per  week  to  teaching;  I  do  not  instruct  in  any 
other  subject. 

121.  Mathematics  and  physics,  also  lectures  on  general  astronomy  and  some 
on  physical  geography, 

123.  Average  eighteen ;  none. 

124.  Graduate  of  the  University  of  Virginia,  with  degrees  of  bachelor  of 
science,  civil  engineer,  mining  engineer ;  two  years  a  student  of  mathematics  in 
Cambridge  University,  England,  and  fourteen  months  at  Gottingen,  Germany  ; 
seventeen  hours  per  week  (mathematics  and  astronomy).  Assistant  Prof.  C.  E. 
Kilbourne,  graduate  of  United  States  Military  Academy,  teaches  ten  hours  per 
week,  and  Assistant  Prof.  G.  W.  McCoard  (Bethany  College,  W.  Va.)  teaches 
eighteen  hours  iJer  week. 

125.  One  and  one-half  years  in  university,  besides  work  done  privately. 
Mathematics,  twelve  hours  per  week  ;  two  hours  in  civil  engineering  and 
astronomy.  , 

127.  I  teach  mathematics  from  fifteen  to  twenty  hours  per  week,  and  French 
from  five  to  ten  hours. 

128.  Aside  from  regular  college  and  private  work,  a  year's  partial  work  at 
Harvard.  Teach  fifteen  hours  a  week.  No  other  subject,  except  Bible,  one 
hour  a  week, 

129.  From  ten  to  fifteen  hours  ;  astronomy,  surveying,  and  bridge  construc- 
tion. 

130.  Four  years,  seventeen  to  eighteen  hours;  astronomy  and  elementary 
mechanics  are  included  in  the  seventeen  or  eighteen  hours. 

131.  Post-graduate  student  two  years  at  Massachusetts  Agricultural  College 
and  Johns  Hopkins  University.  Teach  twelve  to  sixteen  hours  per  week. 
Astronomy  also. 

133.  No  special  training  aside  from  a  regular  college  course  and  private  study ; 
ten  hours ;  mechanics,  physics,  astronomy. 

134.  Teach  three  hours  each  day ;  surveying  and  mechanics. 

'    135.  One  year,  after  graduation  from  college  ;  eleven  hours  per  week ;  astron- 
omy. 

136.  Graduated  with  honors  at  Girton  College,  Cambridge,  1880 ;  four  years' 
subsequent  residence  and  attendance  at  Professor  Cayley's  lectures ;  ten  oi 
eleven  hours  per  week ;  no  other  subject. 

139.  Amount  of  time  occupied  in  teaching  is  regularly  eighteen  hours,  often 
increased  by  extra  work.    I  teach  no  other  subject. 

'     140.  Prof.  Isaac  Sharpless,  L.  B,  (Harvard);  seven  hours;  none.    Prof.  Frank 
Morley,  A.  M.  and  eighth  wrangler  of  Cambridge;  fourteen  hoars;  none. 


MATHEMATICAL  TEACHING  AT  THE  PRESENT  TIME.   349 

State  time  of  your  sjpecial  preparation  for  teaching  mathematics,  numler  of  hours  you  teach 
per  weeh,  and  what  other  subjects  you  feac/i— Continued, 

142.  Usual  college  course  ;  twelve  hours  per  week ;  Latin. 

143.  A  college  course  and  all  tlie  mathematics,  both  regular  aad  extra,  I  could 
crowd  into  it.  I  teach  now  thirteen  to  fifteen  hours  per  week.  No  other  sub- 
ject. 

144.  Algebra,  five  times;  geometry,  five;  trigonometry,  three;  analytical 
geometry,  three ;  calculus,  three ;  analytical  mechanics,  three ;  trigonometry 
and  surveying  by  an  assistant. 

145.  Five  years ;  thirteen  hours  per  week ;  no  other  subject. 

147.  I  have  never  taken  a  special  course  in  mathematics,  but  have  studied 
advanced  works  to  some  extent.  I  teach  mathematics  four  hours  a  day.  I  have 
in  charge  vocal  music,  which  takes  forty  minutes  a  day. 

148.  I  teach  (personally)  twenty  to  thirty  hours  a  week.  About  one-fourth 
of  this  time  is  occupied  with  mathematics,  the  other  three-fourths  with  mechan- 
ics and  civil  engineering. 

.  149.  I  teach  no  other  subjects.    I  teach  twenty  hours  per  week. 

150.  Twenty -five  hours  per  week ;  none. 

151.  No  special  preparation ;  twenty  hours  per  week.  I  teach  no  other  stud- 
ies. 

153.  I  devoted  twelve  years  to  my  special  preparation  for  teaching  mathe- 
matics. 

154.  I  am  a  student  of  mathematics ;  have  charge  of  the  department,  but 
only  teach  the  higher  classes. 

155.  I  first  took  a  college  course  in  mathematics.  After  this  I  spent  several 
years  in  post-graduate  work.  I  teach  eighteen  hours  per  week,  and  I  teach  no 
other  subjects. 

156.  About  seventeen  hours  per  week ;  no  other  subject, 

157.  I  have  a  course  in  civil  engineering.  I  also  have  charge  of  a  commercial 
course. 

^  158.  Number  of  hours  given  above  one  and  a  half  at  thirteen  per  week.  But 
this  does  not  include  office  hours  of  myself  and  assistant  for  meeting  students 
and  giving  explanation.  My  assistant  does  not  teach,  but  simply  keeps  office 
hours  for  consultation  and  solution  and  explanation  of  difficulties  in  lectures  or 
assigned  work.  I  will  add  that  the  under-gradTiate  course  in  pure  mathematics 
la  the  most  extensive  and  thorough  one  given  in  any  university  in  the  United 
States. 

159.  With  assistant,  twenty-four  hours  per  week ;  modern  languages  (French 
and  German). 

160.  Twenty  hours  per  week ;  civil  engineering. 

161.  I  teach  mathematics  four  hours  each  day,  and  also  teach  physics  and 
chemistry. 

163.  Sixteen  hours  per  week ;  eighteen  hours ;  French,  Latin,  history. 

164.  About  twenty-five  hours  per  week. 

166.  We  give  each  class  one  hour  each  day.    Any  subject  in  the  course. 

165.  Twenty  years'  experience  in  the  mathematical  class-room ;  about  ten 
hours  per  week  in  mathematics,  and  five  hours  per  week  in  physics. 

168.  The  number  of  hours  of  teaching  varies  from  five  to  twenty  per  week. 
CoUege  course  and  then  three  years  at  Yalej  physics  and  astronomy. 


350  TEACHING   AND    HISTORY    OF    MATHEMATICS. 

(&)  Normal  Schools. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  per- 
son reporting. 

169 

State  Normal  School 

Jacksonville,  Ala . . . 

C.B.Gibson 

President. 

170 

State  Normal  School 

Florence,  Ala 

J.K.  Powers 

Do. 

L71 

Tnskegee,  Ala 

Pine  Bluff,  Ark 

172 

Branch  Normal  School  of 

J.C.Corbin 

Principal. 

theArkansasIndustrial 

University. 

173 

State  Normal  School 

San  Jos6,  Cal 

R.  S.Holway 

Teacher  in  normal  school. 

174 

State  Normal  School 

Madison,  Dak 

William  F.  Gorrie. 

President. 

175 

Washington  Normal 
School. 

Washington,  D.C... 

E.  S.  Atkinson 

Principal. 

176 

Southern  Illinois  Normal 

Carbondale,  111 

G.  V.  Buchanan . . . 

Teacher  of  mathematics. 

Univeraity. 

177 

Tri-State  Normal  College . 

Angola,  Ind 

L.M.  Sniff 

President. 

178 

State  Normal  School 

Terre  Haute,  Ind  . . . 

Nathan  Ne why ... 

Professor  of  mathematics. 

17D 

State  Normal  School 

Cedar  Falls,  Iowa... 

D.S.Wright 

Do. 

180 

State  Normal  School 

Emporia,  Kans 

M.A.Bailey 

Do. 

181 

€tate  Normal  School 

Gorham,  Me 

W.J.Corthett.... 

Principal. 

1S2 

State  Normal  School 

Baltimore,  Md 

George  L.Smith  .. 

Professor  of  mathematics. 

183 

State  Normal  School 

Westfield,  Mass 

J.  C.  Greenough  .. 

Principal. 

184 

State  Normal  School 

Worcester,  Mass 

E.H.PtUssell 

Do. 

185 

luka  Normal  Institute 

luka.  Miss 

E.L.Sherwood  ... 

Professor  of  natural   sci- 

ence and  mathematics. 

186 

State  Normal  School 

Warrenshurg,  Mo. . . 

George  H.  Howe . . 

Professor  of  mathematics. 

187 

North- Western  Normal 

Stanherry,  Mo 

A.  Moore 

School. 

188 

State  Normal  School 

Kirksvillo,  Mo 

J.I.  Nelson 

Professor  of  mathematics. 

189 

Fremont  Normal  School . . 

Fremont,  Nehr 

W.  H.  Clemmons.. 

President. 

190 

Normal  College  of  New 

Now  York,  N.  Y . . . . 

J.A.Gilbt 

Professor  of  mathematics 

York. 

and  physics. 

191 

State  Normal  and  Train- 

Genesee, N.Y 

E.  A.Waterbury.. 

Professor  of  higher  math- 

ing School. 

ematics  and  methods  in 
arithmetic. 

192 

State  Normal  and  Train- 
ing School. 

Cortland,  N.Y 

D.E.Smith 

Professor  of  mathematics. 

193 

State  Normal  School 

08wego,N.  Y 

W.G.Eappleye... 

Teacher  of  mathematics. 

194 

State  Normal  and  Train- 

New Paltz,  N.  Y  . . . . 

F.  S.Capeu 

Principal. 

ing  School. 

195 

State  Normal  School 

Albany,  N.Y 

E.P.Waterbury.. 

Do. 

196 

State  Normal  School 

Buifalo,N.Y 

J.  M.  Cassedy 

Do. 

197 

State   Colored   Normal 
School. 

Plymouth,  N.C 

H.C.Crosby 

Do. 

198 

Normal  Training  School . . 

Cleveland,  Ohio 

Ellen  E.Keveley.. 

Do. 

199 

North- Western  Normal 

Wauseon,  Ohio 

J.H.Diebel 

Instructor  in  mathematics. 

and  Collegiate  Institute. 

2O0 

State  Normal  School 

Ashland,  Oregon 

J.  S.  Sweet 

President. 

201 

Drain  Academy  and  State 

Drain,  Oregon 

W.C.Hawley 

Do. 

Norma!. 

202 

State  Normal  School 

Bloomsbnrg,  Pa 

G.E.Wilbur 

Professor  of  higher  mathe- 
matics and  history. 

?m 

State  Normal  School 

Philadelphia     Normal 

Clarion,  Pa 

J.H.  Apple 

G.W.Fetter 

Professor  of  mathematics. 

204 

Philadelphia,  Pa  — 

Principal. 

School. 

205 

Cumherland  Valley  State 

Shippensburg,  Pa. . . 

E.H.Bugbee 

Teacher  of  mathematica. 

Normal  School. 

MATHEMATICAL   TEACHING   AT   THE    PEESENT    TIME. 
(5)  NoRMAJL  Schools— Continued. 


351 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  per- 
son reporting. 

206 
207 

208 

State  Normal  School 

Central    State  Normal 

School. 
Eichmond  Normal  School. 

State  Normal  School 

State  Normal  School 

State  Normal  School . . 

State  Normal  School 

State  Normal  School 

"West  Chester, Pa... 
Lock  Haven,  Pa 

Kichmond,  Va 

Fairmont,  W.  Va 

FarmviUe,  Va 

Johnson,  Vt 

Oshkosh,  Wis 

Eiver  Falls,  Wis 

D.M.  Sensonig 

0."W.Kitchell.... 

S.T.  Beach 

C.  A.Sipe 

Professor  of  mathematics. 
Instructor  in  mathematics. 

Principal. 
To. 

210 
211 
212 

Celestia  S.  Parish. 
A.  H.  Campbell  . . . 
E.F. 'Webster 

Teacher  of  mathematics. 

Principal. 

Teacher  of  mathematics. 

State  tim3  of  your  special  preparation  for  teaching  matJiematics,  numher  of  hours  you 
teach  per  week,  and  what  other  subjects  you  teach. 

169.  Four  years ;  twenty  hours ;  physics,  chemistry,  and  astronomy. 

170.  Two  years;  ten  hours;  no  other  subject. 

171.  Twenty  hours ;  reading. 

172.  I  personally  teach,  at  present,  one  class  each  in  algehra,  arithmetic,  and 
geometry,  for  five  days  in  the  week,  one  in  natural  philosophy 

173.  Twenty  hours;  no  other  subject. 

174.  Algebra  five  hours ;  geometry,  Latin,  zoology,  history  of  education. 

176.  Eegular  course  in  university ;  twenty-five  hours ;  no  other  subject. 

177.  Twenty  hours  ;  no  other  subject. 

179.  Twenty  hours;  no  other  subject. 

180.  Twenty  years;  twenty-four  hours ;  no  other  subject. 

182.  Drawing  and  physics. 

183.  The  teacher  of  mathematics  has  ten  hours  ;  physiology. 

185.  Two  years ;  sis  hours ;   natural  science,   history,  rhetoric,  and  book- 
keeping. 

186.  Six  classes  per  day,  forty  minutes  each,  five  days  per  week ;  no  other 
subject. 

188.  Five  hours  per  day ;  astronomy  one  hour  per  day. 
169.  Fifteen  hours. 

190.  Twelve  hours  ;  chemistry,  physics. 

191.  Have  taught  mathematics  almost  exclusively  for'  seventeen  years,  and 
principally  for  twenty-five  years  ;  thirty  hours. 

192.  Election  of  all  mathematics  I  could  get  in  college  course ;  eighteen  and 
three-fourths  hours  ;  class  on  school  law, 

193.  Graduate  of  Cornell ;  twenty-five  hours ;  none. 

195.  Two  years;  twenty  hours;  none. 

196.  Mathematical  course  at  Dartmouth  ;  fifteen  hours ;  astronomy. 

197.  Ten  hours ;  physiology,  history,  moral  science,  and  English  literature. 

198.  Teachers  of  mathematics  not  specialists. 

199.  Six  hours  per  day ;  no  other  subject. 

200.  Twelve  hours ;    book-keeping,  philosophy,  psychology,  art  of  school 
management. 

201.  Seven  hours  jier  week. 

202.  Four  years;   twenty -five  periods,  forty-five  minutes  each,  per  week; 
civil  government. 

203.  Classical  college  course  ;  twenty-five  hours ;  no  other  subject, 

204.  Theory  and  practice  of  teaching  and  school  government 
^05.  Twenty-two  hours ;  no  other  subject. 


352  TEACHING   AND   HISTORY   OF  MATHEMATICS. 

State  time  of  your  special  jyreparation  for  teaching  mathematics,  numier  of  hours  you 
teach  per  weeTc,  and  what  other  subjects  you  teach — Continued. 
■  206.  Graduate  of  both  elementary  and  scientific  courses  of  Millersville  State 
Normal  School,  Pennsylvania  ;  have  taught  for  twenty  years  scarcely  anything 
hut  mathematics  in  throe  of  the  normal  schools  in  Pennsylvania.  Am  author 
of  Numbers  Symbolized,  an  elementary  algebra,  and  have  in  press  Numbers 
Universalized ;  thirty  hours. 

207.  Twenty  hours ;  Latin. 

208.  Algebra,  three  hours ;  physics,  physical  geography,  rhetoric,  Latin,  etc. 

210.  Private  study  at  intervals  for  four  years ;  twenty-five  hours ;  no  other 
subject. 

211.  Usual  course  in  academy,  normal  school,  and  college. 

212.  None,  except  that  spent  in  my  regular  course  in  the  normal  school ;  fif- 
teen hours ;  no  other  branches. 

Are  students  entering  yoiir  institution  thorough  in  thematheynatics  required  for  admission? 
Of  the  forty-five  reports  received  from  normal  schools  three  or  four  give  no  reply  to 
this ;  all  others  answer  no,  excepting  the  institutions  numbered  175,  "  generally  so ; " 
176,  " reasonably  so ; "  180,  ''fairly  so;"  190  and  194,  "yes." 

What  are  the  requirements  in  mathematics  for  admission  ? 

Number  169  reports,  arithmetic  and  elementary  algebra ;  190,  arithmetic  and  a  little 
geometry ;  all  others  require  only  arithmetic,  generally  not  the  whole  of  it,  but 
through  fractions  and  the  simpler  cases  of  percentage.  Number  197  says,  "funda- 
mental rules  of  arithmetic ;  "  204 says,  "fractions  and  percentage."  A  few  institu- 
tions admit  all  who  apply,  without  examination  in  mathematics. 

Js  the  meh'ic  system  taught  ? 

All  answered  in  the  afifirmative,  excepting  those  bearing  the  numbers  171, 185, 190, 
208. 

Which  is  taught  first,  algebra  or  geometry? 
All  answered  "  algebra,  "  excepting  numbers  181, 183, 190,210,211. 
Numbers  181, 183,210  take  up  geometry  first. 
Numbers  190, 211  teach  both  together. 

Sow  far  do  you  proceed  in  the  one  before  taking  up  the  other? 

The  following  carry  students  first  through  a  full  course  of  elementary  algebra: 
169,  170, 174, 180, 184, 185, 189, 201, 206, 208, 209. 

The  following,  through  quadratics :  173, 178, 197. 

The  following,  to  radicals :  193, 196, 212. 

The  following,  to  quadratics :  192, 195, 200, 204. 

The  following,  through  fractions :  188, 207. 

Institution  181  finishes  plane  geometry  before  taking  up  algebra;  183  gives  one 
term  of  geometry  before  algebra;  210  observes  the  following  order:  (1)  A  course  in 
form;  (2)  Rudiments  of  algebra  ;  (3)  Simple  geometric  theorems  and  constructions; 
(4)  More  difficult  algebra ;  (5)  More  difficult  geometry. 
Are  percentage  and  its  applications  taught  before  the  rudiments  of  algebra  or  after? 

All  who  answered  said  "  before, "  except  the  following— 192,  194,  210,  that  said 
"  after ,"  though  some  of  the  simplest  parts  of  percentage  were  taught  before;  184 
and  206  said  that  both  were  taught  together. 

The  mathematical  course  in  the  normal  schools  generally  embraces  a  somewhat  thor- 
ough study  of  arithmetic,  the  study  of  algebra  and  geometry,  and  usually  a  little 
trigonometry. 


MATHEMATICAL   TEACHING   AT   THE   PRESENT   TIME.         353 
(c)  Academies,  Institutes,  and  High  Schools. 


Name  of  institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  per- 
son reporting. 

?tu 

Toroli's  Institnte  for  Boys 

Mobile,  Ala 

A.  Toroli 

Principal. 
Superintendent  of  schools. 

215 

Public  High  School 

Birmingham,  Ala  . . . 

J.  H.  Phillips 

216 

University  High  School  . . 

Tuscaloosa,  Ala 

W.H.Verner 

Principal. 

217 

Marianna  Institute 

Marianna,  Ark 

F.A.Futrall 

'1R 

Kogers,  Ark 

Oakland,  Cal 

J.  W.  Scroggs 

Geo.  C.  Edwards  .. 

Principal. 

Teacher  of  mathematics. 

219 

Hopkins  Academy 

220 

St  Matthew's  Hall 

San  Mateo,  Cal 

H. D. Kobinson  ... 

Tutor  of  mathematics. 

221 

Boys'  High  School 

San  Francisco, Cal.. 

W.N.Bush 

Head-teacher  of  mathemat- 
ical department. 

222 

Los  Angeles  High  School. 

Los  Angeles,  Cal.... 

F.A.Dunham  .... 

Assistant  teacher. 

223 

Girls'  High  School 

San  Francisco,  Cal.. 

Fidelia  Jewett 

Head  of  department  of 
mathematics. 

224 

Oakland  High  School 

Oakland,  Cal 

S.  A.  Chambers    . . 

Teacher  of  mathematics. 

99  i^ 

Hi"h  School 

Colorado  Springs, 
Colo. 

Harriet  Winfield 

Teacher  of  mathematica 
and  science. 

',?fi 

School  for  Boys  ...... . 

Stamford,  Conn 

H.M.King 

M.H.Smith 

Principal. 
Do. 

227 

Connecticut  Literary  In- 

Suffield, Conn 

stitution. 

228 

Public  High  School 

New  Britain,  Conn. . 

John  H.Peck 

Do. 

229 

Sioux  Falls  High  School . . 

Sioux  Falls,  Dak.... 

Anna  Emerson  . . . 

Assistant  high   school 
teacher. 

230 

Washington  High  School. 

Washington,  D.C... 

Charlotte  Smith  .. 

Teacher  of  mathematics. 

231 

Columbian   College  Pre- 

....do  

H.  L.  Hodgkins  . . . 

Instructor  in  mathematics. 

paratory  School. 

'>■},'> 

C.E. Little 

PrincipaL 
Do. 

233 

Academy    of   Kichmond 

Augusta,  Ga 

C.H.Withrow.... 

County. 

234 

Chicago,  111 

Hyde  Park,  HI 

I.  W.Allen 

235 

Public  High  School 

W.H.Eeny 

Principal. 

236 

North  Division  High 
School. 

Chicago,  111 — . . . 

O.LWestcott 

Do. 

?37 

West  Division  High 
School. 

....do 

G.P.Welles 

Do. 

238 

Peoria  High  School 

Joliet  High  School 

South  Division  High 
School. 

Peoria  HI........... 

G.  E.  Knepper 

Do. 

239 

Joliet,  HI 

Do. 

210 

Chicago,  111 

J.  Slocum ...... 

Do. 

241 

Jennings  Seminary 

Aurora,  HI 

J.E.Ad.am3 

Science  and  higher  ma,the- 
matica. 

2*2 

Hi^h  School 

Urbana,  HI . 

J.W.Hays 

Superintendent  of  schools. 
Principal. 

243 

Koanoke  Classical  Semi- 

Koanoke, Ind 

D.N.Howe 

nary. 

244 

Central   Grammar   High 
School. 

Fort  Wayne, Ind  ..- 

Chester  L.  Lane  . . 

Do. 

245 

Public  High  School 

Crawfordsville,  Ind . 

T.H.Dnnn 

Superintendent   of  city 
schools. 

246 

Indianapolis  High  School. 

Indianapolis, Ind  ... 

W.W.Grant 

Principal. 

247 

Westfield,  Ind 

Indianapolis,  Ind  . . . 

M.E.Cox 

Do. 

248 

Indianapolis    Classical 

T.L.S6wall 

Do. 

School  for  Boys. 

249 

Indianapolis  Classical 
School  for  Girls. 

.do 

T.  L.  Sewall,  Mary 
W.  Sewall. 

Principals. 

250 

New  Hope  Female  Acad- 
emy. 

Oak  Lodge,  Choctaw 
Nation,  Ind.  T. 

A.Grifiath 

Suporintendonu. 

SSl—No.  3 23 


354  TEACHING   AND   HISTOEY   OP  MATHEMATICS. 

(c)  Academies,  Institutes,  and  High  Schools— Continued. 


Name  of  institution. 


Location. 


Name  of  person 
reporting. 


Title  or  position  of  per- 
son reporting. 


High  School 

High  School 

Iowa  City  Academy.  — 

High  School 

High  School 

High  School 

High  School 

High  School 

New  Orleans  Seminary 
Girls' High  School 


Madawaska   Training 

School. 
Franklin  Female  College  . 

High  School 

High  School 

Fryeburg  Academy 

High  School 

McDonogh  School 

Washington  County  Male 

High  School. 
Centreville  Academy  and 

High  School. 

Friends'  Academy 

HaverhillTraining  School 
Mount  Hermon  School  — 

C  ushing  Academy 


Prospect  High  School. 


High  School ., 
Eaton  School 


Powder  Point  School  — 

Admiral  Sir  Isaac  Coffin's 
Lancastrian  School. 

Wheaton  Female  Semi- 
nary. 

Lawrence  Academy , 

Smith  Academy 

Partridge  Academy , 

High  School , 

High  School 

Hanover  Academy 

Lynn  High  School 

Bristol  Academy 

High  School 

Nichols  Academy 

High  School 

High  School 

S  a  w  i  n  Academy  and 
Pawse  High  SchooL 


Davenport,  Iowa. . 
Dos  Moines,  Iowa. 
Iowa  City,  Iowa  . . 
Burlington,  Iowa  . 
Davenport,  Iowa. . 

Topeka,  Kans 

Ottawa,  Kans 

Paducab,  Ky 

New  Orleans, La.. 
...do  


Augusta,  Me . 


Topsham,  Me 

Saco,  Me 

Bath,  Me 

Fryeburg,  Me 

Portland,  Me 

McDonogh,  Md  . . 
Hagerstown,  Md . 


Centreville,  Md. 


New  Bedford,  Mass. 

Haverhill,  Mass 

Mount  Hermon, 

Mass. 
Ashburnham,  Mass 

Greenfield,  Mass. . . 

New  Bedford,  Mass 
Middlebor  ough, 
Mass. 

Duxbury,  Ma^ 

Nantucket,  Mass  .. 


Norton,  Mass 

Falmouth,  Mass  . 
Hatfield,  Mass  ... 
Duxbury,  Mass.. 
Peabody,  Mass... 

Salem,  Mass 

Hanover,  Mass . . . 

Lynn,  Mass 

Taunton,  Mass... 
Amherst,  Mass  - . 

Dudley,  Mass 

Haverhill,  Mass  . 
Fitchburg,  Mass. 
Sherborn,  Mass . . 


T.E.Stratton.... 

J.  F.  Gowdy 

M.E.  Tripp 

E.Poppe 

F.E.  Stratton.... 
J.  E.  "Williamson  . 

G.  I.  Harvey 

A.  H.  Beals 

.L.  G.  Atkinson... 
M.C.Cusack  .... 


Vetal  Cye. 


D.L.  Smith 

L.  M.  Chad  wick  . 

H.E.Cole 

M.  E.  Bussell  . . . 

A.  E.Chase 

D.C.Lyle 

G.C.Pearson  ... 


A.  G.  Harley . 


G.B.Dodge... 
C.  A.  Newton . 
H.  E.Sawyer., 


F.D.Lane..., 
Ida  F.  Foster . 


R.  G.  Huling . 
A.A.Eaton. , 


F.  B.  Knapp 
E.B.Fox  ... 


S.  L.  Dawes . 


S.A.Holton  .... 

S.L.  Cutler 

C.F.Jacobs 

C.A.Holbrook.. 
A.  L.  Goodrich.. 
A.  P.  AverUl — 
William  Fuller  . 
"William  F.Palmer 
S.  A.  Sherman  . . 

E.G.Clark 

Clarence  E.Kelley 
H."W.Kjttredge.. 
W.F.Gregory.. 


Principal. 

Teacher  of  mathematica. 

Do. 
Principal. 

Do. 

Do. 
Superintendent. 

Do. 
President. 
Department  of  mathemat- 


Principal. 
Assistant  teacher. 
Principal. 
Assistant. 
Principal. 

Principal. 

Principal. 

Assistant  teacher. 

Principal. 

Superintendent. 

Instructor  in  mathematics 

and  German. 
Teacher    of    science   and 

mathematics. 
Principal. 
Do. 

Do. 
Do. 

Teacher  of  mathematics. 

Principal. 

Do. 

Do. 

Do. 
Master, 
Principal. 

Teacher  of  mathematics. 
Principal. 

Do, 

Do, 

Do. 

Do. 

Do. 


MATHEMATICAL   TEACHING   AT    THE   PRESENT   TIME. 
(o)  Academies,  Institutes,  and  High  Schools— Continued. 


355 


Name  of  Institution. 

Location. 

Name  of  person 
reporting. 

Title  or  position  of  person 
reporting. 

293 

Drnry  High  School 

North  Adams,  Mass . 

Elizabeth  H.  Tal- 
cott. 

First  assistant. 

294 

Charlestown  High  School . 

Boston,  Mass 

J.O.Morris 

(Adeline  L.  Sylves- 
■;    ter 

Head-master. 

^PS 

Girls'  High  School 

....do  

(Emerette  0.  Patch 

WR 

Fablic  Latin  School . . 

do 

G.C.Emery 

"W.  F.Bradbury... 

297 

Cambridge  Latin  School.. 

Cambridge,  Mass  . . . 

Head-master. 

298 

West  RoxburyHigh  School 

Boston,  Mass 

G.C.Mann 

Principal. 

299 

English  and  Classical 
High  SchooL 

Worcester,  Mass 

A.  S.  Roe 

Do. 

sno 

High  School 

Ann  Arbor,  Mich... 

L.  D.  Wines    . 

ematics. 

301 

Michigan  Military  Acad- 

Orchard Lake,  Mich. 

W.H.Butts 

Principal. 

emy. 

30^ 

High  School 

Superintendent    of     city 
schools. 

ROR 

High  School 

Ypsilanti,  Mich 

Grand  Eapids,  Mich 
Kalamazoo,  Mich  . . . 

R.  "W.  Putnam 

Superintendent  of  schools. 
Principal. 
Do. 

30+ 

High  School 

305 

Michigan  Female   Semi- 

Isabella G.French 

nary. 

30fi 

Shattack  School 

Faribault,  Minn 

St.  Cloud,  Minn 

"Wm.  W.  Champ- 

Un. 
C.C.  Schmidt 

307 

Public  High  School 

Superintendent. 

308 

Augsburg  Seminary 

Minneapolis,  Minn. . 

Wilhelm    Petter- 

Instructor. 

309 

Minneanolis  Academy .... 

....do  

E.  D.  Holmes 

Principal. 
Superintendent  of  public 

310 

Public  High  School 

Vicksburg,  Miss 

E.  "W.Wright 

schools. 

311 

Smith  Academy,   Wash- 

St. Louis,  Mo 

E.E.O£Futl 

Teacher  of  mathematics. 

ington  TJniTersity. 

312 

St.  Joseph  High  School ... 

St.  Joseph,  Mo 

C.S.Thacher 

Do. 

313 

Lincoln  High  School 

Lincoln,  Nebr 

S.P.Barrett 

PrincipaL 

314 

Eobinson  Female   Semi- 

Exeter, N.H 

G.N.Cross 

Do, 

nary. 

315 

Simonds  Free  High  School . 

Warner,  N.H 

E.  P.  Barker .,,... 

Do. 

316 

Concord  High  School 

Concord,  N.H 

J.F.Kent 

Do. 

317 

Brewster  Free  Academy., 

"Wolf borough,  N.H. 

E.H.Lord 

Do. 

318 

High  School 

Portsmouth,  N.H... 
Pennington,  N.  J 

Do. 

319 

Pennington  Seminary 

J.R.Hamlon 

"Vice-President. 

320 

Hoboken  Academy 

Hoboken,  N.J 

J.  Sohrenk 

Principal. 

3''1 

High  School 

Newark,  N.J 

Oranse.N.  J 

H.  T.  Dawson 

W.W.  Cutts 

322 

PubHc  High  School 

Principal. 

323 

Newark  Technical  School. 

Newark,  N.J 

H.  T.Dawson 

Instructor  in  mathematics. 

324 

Blair  Presbyterial  Acad- 

Blairstown, N.  J 

J.  H.  Shumaker . . . 

Principal. 

emy. 

"-■ 

325 

Stevens  High  School 

Hoboken,  N.J 

E.  L.  Sevenoak  . . . 

Assistant    principal    and 
professor  of    mathe- 
matics. 

326 

Newark  Academy 

Newark,  N.J 

S.A.Farrand 

Head-master. 

327 

Dearborn  Morgan  School. 

Orange,  N.  J 

D.  A.  Kennedy 

Principal. 

3S8 

Ne-w   Brunswick   High 
School. 

Superintendent  of  schools. 

356  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

(o)  Academies,  Institutes,  and  High  Schools — Continued. 


ITame  of  institution. 


Location. 


Fairfield  Seminary 

College  Grammar  School. 

Cazenovia  Seminary 

Ires  Seminary 

St.  Mary's  Academy 

Adams  Collegiate   Insti- 
tute. 

Grammar  School 

Gonverneur  Seminary  — 
Union  Classical  Institute. 

High  School 

Oxford  Academy 

Brooklyn  Latin  School — 

The  trtica  Academy 

Rome  Free  Academy 

High  School 

High  School 

Buarley  School  for  Girls., 

Friend's  Seminary 

Central  High  School 

School  for  Girls 

Free  Academy , 

Delavan  Academy 

Port  Jervis  Academy 

Tonlcera  High  School 

High  School 

High  School 


Fremont  Institute. 


Bingham  School . 


High  School 

Green  To-n^n  Academy. . . 

High  School 

High  School 

Mission  House  College. . . 
Bishop  Scott  Academy. . . 


Dickinson  Seminary 

Western      Pennsylvania 

Classical  and  Scientific 

Institute. 
Philadelphia  Seminary  for 

Toung  Ladies. 

"Wyoming  Seminary 

Harry  HUlman  Academy. 
"William    Penn    Charter 

School. 

Central  High  School 

High  School 


371  High  School . 

372  High  School , 


Fairfield,  N.T 

Brooklyn,  N.  T 

Cazenovia,  N.  T 

Antwerp,  N.  T 

Troy.N.  Y 

Adams,  N.  T 

Clinton,N,Y 

Gouverneur,  N.  T. . . 
Schenectady,  N.  Y  . . 

Troy.N.Y 

Oxford,  N,Y 

Brooklyn,  N.Y 

irtica,Iir.Y 

Kome,lSr.Y 

Buffalo,  N.Y 

Poughkeepsie,  N.  Y. 
iS^ew  York,  2f.Y..., 

Ne-sv  York,  N.  Y 

Einghamton, 'N.Y  .. 
New  York,  K.  Y  . . . 

Elmira,N".  Y , 

Delhi,K.Y ..., 

Port  Jervis,  IST.  Y  . . . 

Yonkers,  K".  Y 

Albany,  K.  Y 

Syracuse,  N.  Y 

Fremont,  N.  C , 

Bingham   School 

P.O.,N.C. 
HnntorsvUle,  N.  C  . , 
Perrysville,  Ohio  .. 
Zanesville,  Ohio  . . . 

Dayton,  Ohio 

Cleveland,  Ohio 

Portland,  Oreg 

"Williamsport,  Pa  . . 
Mt.  Pleasant,  Pa . . . 


Philadelphia,  Pa . . 

Kingston,  Pa 

Wilkes  Barre,  Pa  . 
PhQadolphia,  Pa . . 


Kame  of  person 
reporting. 


Chester,  Pa 

Titusville,  Pa 

West  Chester,  Pa. 
Scrauton,  Fa 


J.M.Hall 

L.W.Hart 

A.White 

E.M.Wheeler  .. 

John  Hogan 

L.  B.  Woodward 


Isaac  O.  Best 

J.  F.Ferthill 

E.E.Veeder 

J.  P.Worden 

F.  L.  Gamage 

C.Harrison 

G.  C.  Sange 

M.  T.  Scuddor 

M.T.Karnes 

James  Winne 

Jeannette  Fine  . . 
John  M.  Child  . . . 
Fannie  Webster. 

A.  Brackett 

E.T.  Wilson 

W.D.Graves  .... 
JohuM.  Dolph... 

E.  E.Shaw 

J.H.Gilbert 

0.  C.  Kinyon 


W.  Wills 


11.  Bingham. 


Title  or  position  of  person 
reporting. 


W.YAOrr 

J.C. Sample 

W.  M.  Townsend. 

C.B.Stivens   

J.W.Grosshnesch. 
F.  E.  Patterson  . . . 


G.  G.Brower. 
L.  Stephens. . 


Carrie  A.  Bitting. 

E.  B.  Howland  . . . 

E.Scott 

A.D.Gray 

J.  F.  Eoizart 

C.E.Eose 

J.E.Philips 

J.  C.Lange 


Teacher  of  sciences. 

PrincipaL 

Teacher  of  mathematics. 

Principal. 

Teacher  of  classics. 


Principal. 

Superintendent  of  schools. 
Teacher  of  mathematics. 
Professor  of  mathematics. 
Principal. 
Headmaster. 
PrincipaL 

Do. 

Do. 

Do. 
Teacher  of  mathematics. 
Principal. 

Instructor  in  mathematics. 
Principal. 

Teacher  of  mathematics. 
Principal. 
Superintendent. 
Principal. 

Professor  of  mathematics. 
Teacher   of   physics    and 

mathematics. 
Instructor  in  mathematics 

and  Latin. 
Superintendent. 

President. 

Do. 
Principal. 

Do. 
Professor. 

Lieutenant  Colonel— math- 
ematics. 
Teacher  of  mathematics. 
President. 


Librarian. 

Teacher  of  mathematics. 

Principal. 

Teacher  of  mathematics. 

Principal. 

Head  of  mathematical  de- 
partment! 
Teacher  of  mathematics. 
Principal. 


MATHEMATICAL    TEACHING   AT   THE   PRESENT   TIME.         357 
(c)  Academies,  Institutes,  aj^d  High  Schools — Continued. 


373 
374 

375 
376 
377 
378 
379 
380 
381 
382 

383 
384 

385 
386 
387 
388 
389 
330 
391 
392 
393 
394 


■Name  of  Institution. 


High  School 

High  School 

High  School 

Boys'  High  School 

High  School 

Pawtucket  High  School. . 

High  School 

High  School.... 

High  School 

Thetford  (Vt. )  Academy 
and  Boarding  School. 

Brigham  Academy 

Troy  Conference  Acad- 
emy. 

High  School 

Central  Female  Institute 

High  School , 

Thyne  Institute 

High  School 

"West  Virginia  Academy. . 

Male  Academy 

Free  High  School , 

High  School , 

High  School 


Location. 


"Wilkes  Barre, Pa... 

York,  Pa 

Carbondale,  Pa 

Harrisbnrg,  Pa 

Providence,  K.  I 

Pawtucket,  K.  I 

Charleston,  S.  C 

Chattanooga,  Tenu. 

Austin,  Tex 

Thetford,  Vt 

Bakersfleld,  Vt 

Poultney,Vt 

Rutland,  Vt 

GordonsviUe,  Va 

Richmond,  Va 

Chase  City,  Va 

Charleston,  "W.  Va  . . 
Buckhannon,  "W.  Va 
Charlestown,  "W.  Va. 

Sheboygan,  "Wis 

Milwaukee,  "Wis 

Oshkosh,  "Wis 


Name  of  person 
reporting. 


0.  "W.Potter... 

A.  "Wanner 

H.  J.  Hoeckenburg 

J.H."Wert 

D."W.Hoyt 

"W."W.  Curtis 

V.C.  Dibble 

J.B.Cash 

J.B.  Bryant 

E.F.Morse 

F.E.Parlin 

C.H.Dunton 

L.  B.  Folsom 

Jas.  Dinwiddie 

"W.  F.Fox 

J.  H.  Veazey 

M.  K.  McGwigan. . 

W.  Johnson 

E.R.Taylor 

E.G.  Haylett 

G.  "W.  Peckham  . . 
R.  H.  Halsey 


Title  or  position  of  person 
reporting. 


Superintendent  of  sohoola. 
PrincipaL 

Do. 

Do. 

Do. 
Head-master. 
Principal.  > 

Do. 

Do. 
Assistant  teacher. 

Principal. 
Do. 

Do. 
Do. 
Do. 

Superintendent  of  schools. 

Principal. 

Teacher  of  mathematics. 

Principal. 

Do. 

Do. 

Do. 


What  reforms  are  needed  in  the  teaching  of  arWhraetie  ? 

215.  Less  adherence  to  and  dependence  upon  text-books ;  more  thorough 
primary  drill. 

218.  More  easy  examples. 

223.  More  mental  "work,  more  analytical  work,  greater  quickness. 

2iJ5.  Increase  in  number  of  problems  under  each  principle,  decrease  in  nunio 
her  of  "  catch  problems"  ;  more  mental  work. 

229.  There  is  too  much  time  put  on  it  in  all  the  lower  grades. 

232.  More  attention  to  rapidity,  more  every-day  sums. 

237.  Introduction  of  quick  and  labor-saving  methods  in  all  business  methods. 

242.  Better  use  of  mathematical  language;  arithmetic  as  a  deductive  science. 

251.  More  practice  in  rapid  calculation.  Many  of  the  unimportant  rules 
should  bo  scarcely  touched.     My  pupils  waste  energy  by  scattering  too  much. 

255.  A  more  judicious  selection  of  subjects  that  time  be  not  wasted  upon 
non-essentials. 

257.  More  mental  arithmetic. 

262.  Something  to  make  it  more  practical  and  the  student  better  able  to 
apply  it. 

270.  Fundamental  operations  of  arithmetic  only  should  be  taught  before 
algebra. 

274.  Text-books  are  either  so  childish  as  to  give  no  inspiration  to  work 
after  the  primary  grades,  or  so  abstruse  and  dependent  upon  logical  reason- 
ing beyond  a  child's  capacity  as  to  discourage. 

275,  Insistence  upon  accuracy  in  fundamental  operations,  and  alertness  of 
mind  everywhere. 


358  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

What  reforms  are  needed  in  the  teaching  of  arithmetic  f — Continued. 

276,  More  tliorough  work  iu  elementary  rules  and  in  common  and  decimal 
fractions. 

277.  Scholars  are  pushed  ahead  altogether  too  fast,  allowed  to  work  slowly 
and  incorrectly;  should  he  drilled  in  quick  addition,  etc. 

281.  More  attention  to  accuracy,  rapidity,  and  practical  methods. 
283.  It  should  he  taught  as  an  art  rather  than  as  a  science. 
286.  There  should  he  vastly  more  drill  in  fundamental  processes. 

288.  Plenty  of  examples,  more  oral  and  "mental"  work. 

289.  More  practice.  It  seems  to  me  that  the  agitation  for  reducing  tinie  given 
to  arithmetic  is  a  mistake,  though  greater  economy  of  effort  is  possible. 

294.  Fewer  subjects,  more  speed  aod  accuracy  in  computation. 

297.  The  difficulty  (especially  with  female  teachers)  is  too  great  subserv- 
iency to  the  text-book — lach  of  elafiticity  in  accepting  methoda. 

300.  Hire  competent  teachers  only. 

304.  More  mental  arithmetic. 

307.  More  mental  work,  greater  accuracy  and  rapidity.  Scope  of  the  subject 
reduced. 

312.  More  practical  work;  judicious  omission  from  ordinary  text-book;  bet- 
ter development  of  principles. 

314.  More  mental,  less  written  work. 

317.  A  diminution  in  the  number  of  subjects  and  more  independent  work  by 
the  pupil. 

322.  Particular  attention  to  thoroughness,  and  abundant  practice  on  funda- 
mental rules  and  business  methods,  with  the  omission  of  some  rules  and 
methods  formerly  deemed  essential. 

331.  Keep  the  keys  out  of  the  way  and  analyze  every  problem. 

335.  To  return  to  the  old  custom  of  making  the  pupil  do  more  thinking. 
There  are  too  many  helps  and  too  much  "mince-meat.", 

338.  Many. 

340.  More  philosophy. 

341.  Return  to  mental  arithmetic,  now  sadly  neglected.  More  attention  to 
analysis,  less  to  ingenious  devices. 

344.  Do  not  permit  primary  teacher  to  ube  a  figure  in  presence  of  children  till 
they  know  everything  about  numbers  one  to  ten. 

346.  The  use  of  the  Griihe  method  with  beginners,  of  denominate  numbers  be- 
fore abstract,  the  expansion  of  the  method  of  analysis  in  solving  problems 
usually  assigned  to  proportion. 

352.  A  method  that  will  shorten  the  time,  give  the  pupils  the  essentials 
thoroughly.  This  will  come,  I  believe,  only  through  the  experiments  in  indus- 
trial education. 

353.  More  simplicity,  less  aiming  to  puzzle,  less  work  that  is  wholly  theo- 
retical. 

359.  Brief  methods  of  calculation  should  be  insisted  on,  also  independence. 

370.  Less  of  it,  in  much  less  time  than  is  noio  given  to  it  {Superintendent  of 
Schools). 

382.  More  attention  to  mental  arithmetic. 

386.  The  use  of  such  books  as  Colburn's  or  Venaole's  Mental  Arithmetic  thor- 
oughly at  first ;  and  the  rejection  of  such  methods  as  have  recently  been  in- 
jected into  the  new  Colburn's  Mental  Arithmetic.  The  public  schools  are 
teaching  for  show. 

389.  Books  without  answers  are  needed. 

392.  We  should  not  go  too  far  in  seeking  to  make  all  divisions  in  arithmetio 
practical.    Discipline  must  be  held  in  mind. 


MATHEMATICAL   TEACHINa  AT   THE   PRESENT   TIME.         359 

To  what  extent  are  models  used  in  teaching  geometry  ? 

The  following  reported  that  models  were  not  used:  216,219,220,226,238,239,246, 
253, 256, 257, 263, 266, 267, 277, 278, 288, 299, 300,  302, 316,  334, 352, 370, 378, 384, 393. 

The  following  reported  "occasionally,"  "not  much,"  "  very  little " :  217,218,222, 
231,  233,  236,  240,  241,  242, 243, 244, 245, 251, 255, 258, 265, 269, 276, 282, 293, 294, 295, 296, 
307, 309, 318, 324, 326, 332,  335,  337,  338, 339,  341,  348,  350,  351, 354, 356, 358, 360, 364, 366, 
367,  372,  375,  385,  387,390,  391. 

Nearly  all  the  remaining  reports  stated  that  models  were  used,  specifying,  in  many 
cases,  that  they  were  found  particularly  serviceable  in  teaching  solid  and  spherical 
geometry. 

Those  reports  which  stated  that  the  models  were  made  by  the  pupils  themselves 
were  classified  with  the  group  "using  models."  To  teach  plane  geometry  to  very 
young  students,  or  solid  and  spherical  geometry  to  students  of  any  grade,  without 
the  aid  of  models,  is  a  great  mistake. 

To  what  extent  and  icith  what  success  original  exercises  ? 

All,  except  about  two  dozen,  reported  that  original  exercises  were  frequently  used, 
with  good  success.  Some  said  that  one-sixth  of  the  time  allotted  to  geometry 
was  devoted  to  them,  others  said  one-half  of  the  time;  but  the  large  majority  of 
those  specifying  the  relative  amount  of  time  given  to  such  work  answered  one-fourth. 
Several  reporters  took  occasion  to  say  that  the  teaching  of  geometry  without  intro- 
ducing original  exercises  was  necessarily  more  or  less  of  a  failure. 

Is  the  metric  system  taught  ? 

Nearly  every  report  showed  that  this  is  taught,  though  in  many  schools  but  little 
attention  is  given  to  it.  We  observed  only  one  instance  in  which  it  was  "  dropped," 
after  having  been  taught  for  some  years.  How  long  will  it  be  before  this  country 
will  wheel  in  line  with  the  leading  European  nations  and  adopt  this  system  to  the 
exclusion  of  the  wretched  systems  now  in  use  among  us  ? 

Which  is  taught  first,  algebra  or  geometry?    Hoiv  far  do  you  proceed  in  the  one  before 

taking  up  the  other  ? 

Excepting  a  number  less  than  a  dozen,  all  answered  that  algebra  was  taught  first. 
The  following  complete  a,  couTae  in  elementary  algebra,  before  taking  up  geometry: 
215,  218,  225,  226, 230,  235,  237,  238,  239, 245, 246,  247,  260,  263, 266, 267,  272, 273, 278, 282, 
283,  284,  289,  290,  292,  294,  298.  299,  301,  307,  314,  316,  318,  321 ,  323, 324,  326, 327, 328, 335, 
337, 338,  344,  345,  353,  354,  355,  359,  360,  369,  370, 372,  376, 377,  380, 382,  383,  385,  387,  389, 
394. 

The  followin  g  take  up  geometry  after  having  carried  the  student  through  quadratics : 
214,  223,  236,  244,  252,  255,  256,  258,  263,  274,  275,  280,  286,  291,  295, 297, 300, 303, 305, 306, 
311,  317,  334,  336,  339,  343,  350,  351,  356,  363,  378,  381,  384, 391. 

The  following,  after  having  carried  the  student  to  quadratics :  216,  217,  233,  250, 
251,  257,  264,  302,  357,  367,  373,  386. 

Through  radicals :  228,  243,  262,  322. 

Through  equations  :  224,  268,  330. 

To  simple  equations :  219, 2Sl,  288,  374,  379. 

Through  factoring  :  276,  325. 

Through  L.  C.  M.  and  G.  C.  D.:  220. 

To  fractions  :  232, 248,  249. 

To  involution  :  229. 

Of  those  who  take  up  geometry  before  algebra,  222  teaches  Hill's  Geometry  lor  Be- 
ginners, 234  teaches  the  simpler  parts  of  geometry,  242  teaches  mathematical  draw- 
ing, involving  about  sixty  geometric  problems  (without  demonstrations),  315  teaches 
geometry  one  year,  293  observes  the  following  order  of  studies :  (1)  Beginning  geome- 
try; (2)  algebra;  (3)  geometry. 

In  the  two  institutions,  269  and  270,  algebra  and  geometry  are  taught  together.  la 
this  scheme  not  worthy  of  more  extended  trial  ? 


360  TEACHING   AND   HISTORY    OF   MATHEMATICS. 

Are  percentage  and  its  applications  taught  lefore  the  rudiments  of  algebra  or  after  f 

Nearly  all  replied  that  it  was  taught  "  before." 

The  following  answered  that  in  their  institution  it  was  taught  "after":  217,  224 
(arithmetic  being  reviewed  with  aid  of  algebra),  238,  239,  252,  266  (review),340,  347, 
355,  360. 

In  most,  if  not  all  these  cases,  the  elements  of  percentage  had  been  taught  to  the 
pupil,  before  he  entered  the  institution. 

In  325  the  two  subjects  are  taught  "  together." 

Is  it  not  desirable  to  introduce  the  rudiments  of  algebra  earlier  than  has  been  the 
custom  in  most  of  our  schools? 

Are  pupils  permitted  to  use  "  answer-looks  "  in  arithmetic  and  algehra  f 

"  Yes,"  "  yes,  but  not  encouraged" :  214, 216, 217, 218, 219, 226  (with  younger  classes), 
228,  230,  235,  238,  239  (in  arithmetic),  243,  245,  246,  247,  248,  249,  250,  251,  255,  257, 
260, 276,  277, 281, 289,  295,  302,  328,  330,  334,  335,  337,  339,  340,  342,  344,  347,  350, 356, 357, 
359,  369  (in  algebra,  but  not  in  arithmetic),  370,  .371,  373,  374,  380,  381,  386,  387,  388, 
389,391,  394. 

"  No  " :  215, 220,  221, 222,  224,  226  (with  older  classes),  230,  237, 239  (in  algebra,  but 
not  in  arithmetic),  244,  256  (in  algebra),  264,  278,  286,287,  288,  290,  291,  293,  299,  304, 
305,  310,  314, 315,  317,  318,  321,  323,  327, 331,  338,  343,  348,  349,  351, 360,  367,  372  (in  alge- 
bra), 375,  385,  393. 

"Some of  the  answers:"  214,215,221,  223,  224,  225,  228,  235,  236,  237,  238,  239,  240, 
242.244.245. 

Are  students  entering  your  institution  thorough  in  the  mathematics  required  for  admission  ? 

Some  of  the  institutions,  especially  academies  and  institutes,  have  no  require- 
ments for  admission.  In  the  great  majority  of  reports  there  was  a  general  complaint 
that  students  were  "not"  well  prepared  or  "by  no  means"  well  prepared  in  the 
requisites  for  admission. 

The  following  answered  "  yes,"  "  fairly  so : "  214, 220, 221, 223, 224, 227, 228, 235, 237, 
238, 245, 254, 266, 268, 293, 311, 322, 337, 352, 359, 390,  394. 

What  are  the  requirements  in  mathematics  for  admission  to  the  institution  f 

"Practical  arithmetic,"  "common  school  arithmetic,"  was  the  reply  given  by  one 
hundred  and  fourteen  institutions. 

"  Cube  root  in  arithmetic  and  equations  of  the  second  degree  in  algebra,"  217. 

Arithmetic  and  elementary  algebra  :  222, 230, 273, 288, 303, 357. 

Three  books  in  geometry,  Brook's  Algebra,  and  arithmetic,  268. 

Arithmetic  and  algebra  as  far  as  factoring,  370. 

To  ratio  and  proportion  in  Olney's  Practical  Arithmetic,  394. 

Arithmetic  through  percentage,  360. 

Arithmetic  to  percentage:  218,  306, 328, 379, 383. 

Through  fractions  in  arithmetic  :  233, 282,  317. 

Fundamental  roles  in  arithmetic :  269, 356, 391. 


V. 

HISTORICAL  ESSAYS. 


HISTORY  OF  INFINITE  SERIES.* 

The  primary  aim  of  this  paper  is  to  consider  the  views  on  infinite 
series  held  by  American  mathematicians.  But  the  historical  treatment 
of  this  or  any  similar  subject  would  be  meagre  indeed  were  we  to  confine 
our  discussion  to  the  views  held  by  mathematicians  in  this  country. 
We  might  as  well  contemplate  the  growth  of  the  English  language 
without  considering  its  history  in  Great  Britain,  or  study  the  life-history 
of  a  butterfly  without  tracing  its  metamorphic  development  from  the 
chrysalis  and  caterpillar.  A  satisfactory  discussion  of  infinite  series 
makes  it  necessary  that  the  greater  part  of  our  space  be  devoted  to  the 
views  held  by  European  mathematicians. 

Previous  to  the  seventeenth  century  infinite  series  hardly  ever  oc- 
curred in  mathematics  5  but  about  the  time  of  Newton  they  began  to 
assume  a  central  i)osition  in  mathematical  analysis. 

Wallis  and  Mercator  were  then  employing  them  in  the  quadrature  of 
curves.  I^ewton  made  a  most  important  and  far-reaching  contribution 
to  this  subject  by  his  discovery  of  the  binomial  theorem,  which  is  en- 
graved upon  his  tomb  in  Westminster  Abbey.  ^Newton  gave  no  dem- 
onstration of  his  theorem  except  the  veritication  by  multiplication  or 
actual  root  extraction.  The  binomial  formula  is  a  finite  expression 
whenever  the  exponent  of  {a+h)  is  a  positive  whole  number;  but  it  is 
a  series  with  an  infinite  number  of  terms  whenever  the  exponent  is 
negative  or  fractional.  Newton  appears  to  have  considered  his  formula 
to  be  universally  true  for  any  values  of  the  quantities  involved,  no 
matter  whether  the  number  of  terms  in  the  series  be  finite  or  infinite. 

The  binomial  theorem  was  the  earliest  mathematical  discovery  of 
Newton.  Further  developments  on  the  subject  of  infinite  series  were 
brought  forth  by  him  in  later  works.  He  made  extensive  use  of  them 
in  the  quadrature  of  curves.  Infinite  series  came  to  be  looked  upon  as 
a  sort  of  universal  mechanism  upon  which  all  higher  calculations  could 
be  made  to  depend.    Special  methods  of  computation,  such  as  contin- 

*TMs  article  was  read  before  the  New  Orleans  Academy  of  Sciences  in  December, 
1887,  and  printed  in  the  "Papers"  published  by  that  society.  Vol.  I,  No.  2.  Some 
Blight  changes  have  been  introduced  here. 

361 


362  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

ued  fractions,  could  easily  be  reduced,  it  was  said,  to  the  general  method 
of  infinite  series.  It  thus  appears  that  series  were  cultivated  by  the 
early  analysts  with  great  zeal.  They  seem  to  have  placed  perfect  con- 
fidence in  the  universality  of  the  method.  The  mass  of  mathematicians 
never  dreamed  that  the  unrestricted  use  of  infinite  series  was  under- 
mining mathematical  rigor  and  opening  avenues  of  doubt  and  error  j 
they  had  no  idea  that  in  reasoning  by  means  of  series  it  was  necessary 
to  consider  their  convergency  or  divergeticy.  To  show  what  implicit  con- 
fidence was  placed  in  infinite  series  I  shall  quote  a  passage  from  the 
large,  and  in  many  respects  excellent,  history  of  mathematics,  written 
by  the  celebrated  Montucla,  who  flourished  during  the  latter  half  of  the 
eighteenth  century. 

In  Volume  III,  page  272,  he  expresses  the  desirability  of  having  a 
more  rigid  demonstration  of  the  binomial  formula  than  that  given  by 
I^ewton,  so  that  no  rational  being  might  ever  entertain  the  faintest 
doubt  of  its  truth.  Among  the  early  English  mathematicians  there  was 
one  who  did  raise  objections  to  the  binomial  formula,  and  of  him  Mon- 
tucla says:  "Thus  we  have  seen  a  certain  Dr.  Green,  *  *  *  although 
professor  of  physics  at  the  University  of  Cambridge  and  a  colleague  of 
Cotes,  not  only  doubt  it,  but  pretend  that  it  was  false  and  say  he  could 
prove  it  by  examples  badly  applied;  but  it  does  not  appear  that  the  En- 
glish geometers,  not  even  Cotes,  his  colleague,  deigned  to  reply  to  him." 
In  the  light  of  modern  science,  this  passage  ridiculing  Green  is  very  in- 
structive. Time  has  turned  the  tables,  and  the  laugh  is  no  longer  upon 
Green,  but  upon  Montucla  himself.  We  now  wonder  at  the  reckless- 
ness with  which  infinite  series  were  once  used  in  mathematical  reason- 
ing. To  be  sure,  talents  of  the  first  order,  such  as  Newton,  Leibnitz, 
Euler,  Clairaut,  D'Alembert,  possessed  too  much  tact  and  intuitive  in- 
sight to  permit  themselves  to  be  dragged  to  the  dangerous  extremes 
and  yawning  precipices  of  error,  toward  which  their  own  imperfect 
theory  of  infinite  series  tended  to  draw  them.  And  yet,  some  of  them 
did  not  escape  blunders.  The  penetrating  and  teeming  mind  of  Euler, 
for  instance,  is  said  to  have  fallen  into  some  glaring  mistakes  by  the 
incautious  use  of  infinite  series. 

Among  the  mathematicians  who,  above  all  others,  made  the  most 
unrestricted  and  reckless  use  of  infinite  series,  were  the  Germans. 
There  flourished  in  Germany  during  the  latter  part  of  the  eighteenth 
century  a  mathematical  school  which  occupied  itself  principally  with 
what  was  termed  "  combinatorial  analysis."  This  analysis  was  culti- 
vated in  Germany  with  singular  and  perfectly  national  i)redilection. 
One  of  the  first  problems  considered  by  them  was  the  extension  of  the 
binomial  formula  to  polynomials,  and  the  devising  of  simple  rules  by 
which  polynomials  could  be  developed  into  series.  The  solution  of 
this  problem  was  followed  by  the  i)roblem  of  "reversion  of  series."  In 
this  connection  a  quotation  from  Gerhardt's  GescMchte  der  Mathematik 
in  Beutscliland  (p.  205)  is  instructive. 


HISTORICAL   ESSAYS.  363 

Says  lie :  "  The  advocates  of  the  combinatorial  analysis  were  of  the 
opinion  that  with  the  complete  solution  of  this  problem  (of  reversion 
of  series)  was  given  also  the  general  solution  of  equations.  But  here 
they  overlooked  an  important  point — the  convergency  or  divergency 
of  the  series  which  was  obtained  for  the  value  of  the  unknown  quantity. 
Modern  analysis  justly  demanded  an  investigation  of  this  point,  inas- 
much as  the  usefulness  of  the  results  is  completely  dependent  upon  it." 
It  thus  appears  that,  through  the  misuse  of  infinite  series,  the  Germans 
were  temporarily  led  to  believe  that  they  had  reached  a  result  which 
mathematicians  had  so  long  but  vainly  striven  to  attain,  namely,  the  al- 
gebraic solution  of  equations  higher  than  the  fourth  degree.  It  will  be 
observed  that  their  method  lacked  generality,  since  it  could  at  best  not 
yield  more  than  one  root  of  an  equation.  But  in  the  determination  of 
this  one  root  the  combinatorial  school  was  deceived.  The  result  was  a 
mere  delusion— a  mirage  produced  by  the  refraction  of  the  rays  of 
reasoning  from  their  true  path  while  passing  through  the  atmosphere 
of  divergent  series. 

We  proceed  now  to  the  farther  consideration  of  the  binomial  theo- 
rem. After  the  time  of  Newton  numerous  proofs  were  given  of  the 
binomial  theorem.  James  Bernoulli  demonstrated  the  case  of  whole 
positive  powers  by  the  application  of  the  theory  of  combinations. 

This  proof  is  excellent,  and  has  retained  its  place  in  school-books  to 
the  present  day.  But  the  general  demonstration  for  the  case  where  the 
exponent  may  be  negative  or  fractional  was  still  wanting.  Maclaurin 
was  among  the  first  to  offer  a  general  demonstration.  Soon  after  his 
followed  a  host  of  proofs,  each  of  which  met  with  objections.  It  is  no 
great  exaggeration  to  say  that  these  early  demonstrations  seemed  to 
satisfy  no  one  excepting  their  own  authors.  Most  celebrated  is  the 
proof  given  by  Euler.  It  is  still  found  in  some  of  our  algebras.  But 
Euler's  proof  has  one  fault  which  is  common  to  nearly  all  that  have 
been  given  of  this  theorem.  It  does  not  consider  the  convergency  of 
the  series.  It  seems  to  me  that  this  fault  is  fatal.  Buler  claims  to  prove 
that  the  binomial  formula  is  generally  true,  but  if  this  formula  is  act- 
ually taken  as  being  universally  true,  then  it  can  be  made  to  lead  to  all 
sorts  of  absurdities.    If,  for  instance,  we  take,  in  (a  -f-  &)",  a  =  1,  &  = 

—  3,  w  =  —  2,  then  we  get  from  the  formula  —  =  oo. 

One  might  think  that  absurdities  of  this  kind  would  have  brought 
about  the  immediate  rejection  of  all  proofs  neglecting  the  tests  of  con- 
vergency, but  this  has  not  been  the  case. 

Another  infinite  series  occupying  a  central  position  in  analysis  is  the 
one  known  to  students  of  calculus  as  Taylor's  theorem.  It  was  discov- 
ered by  Brook.  Taylor  and  published  in  London  in  1715.  One  would 
have  thought  that  the  instant  it  was  proposed,  this  theorem  would  have 
been  hailed  as  the  best  and  most  useful  of  generalizations.  Instead  of 
this  it  remained  quite  unknown  for  over  fifty  years,  till  Lagrange 


364  TEACHING   AND    HISTORY   OP   MATHEMATICS. 

pointed  out  its  power.  In  1772  Lagrange  publislied  a  memoir  in  which 
he  proposed  to  make  Taylor's  theorem  the  foundation  of  the  differential 
calculus.  By  doing  so  he  hoped  to  relieve  the  mind  of  the  difiicult  con- 
ception of  a  limit  upon  which  the  calculus  has  been  built  by  Newton 
and  his  disciples.  The  method  of  limits  was  then  involved  in  philo- 
sophic difficulties  of  a  serious  nature.  It  was  therefore  very  desirable 
that  an  explanation  of  the  fundamental  principles  should  be  given  which 
should  be  so  clear  and  rigorous  as  to  command  immediate  assent.  The 
illustrious  Lagrange  attempted  to  supply  such  an  explanation.  He 
boldly  undertook  to  prove  Taylor's  theorem  by  simple  algebra,  and  then 
to  deduce  the  whole  differential  calculus  from  Taylor's  theorem.  In 
this  way  the  use  of  limits  or  of  infinitely  small  quantities  was  to  be  dis- 
pensed with  entirely.  If  Taylor's  theorem  be  once  absolutely  granted, 
then  undoubtedly  all  the  rest  may  be  made  to  follow  by  processes  which 
are  strictly  rigorous.  But  in  proving  Taylor's  theorem  by  simple  alge- 
bra without  the  use  of  limits  or  of  infinitesimals,  Lagrange  avoided  the 
whirlpool  of  Charybdis  only  to  suffer  wreck  against  the  rocks  of  Scylla. 
The  principles  of  algebra  employed  by  him  in  his  proof  were  those  which 
he  received  from  the  hands  of  Euler,  Maclaurin,  and  Olairaut.  His  proof 
rested  chiefly  upon  the  theory  of  infinite  series.  But  we  have  seen 
that  this  very  theory  was  at  that  time  wanting  in  mathematical  rigor. 
Consequently,  all  conclusions  evolved  from  it  possessed  the  same  defect. 
Though  Lagrange's  method  of  treating  the  calculus  was  at  first  greatly 
applauded,  objections  were  afterward  raised  against  it,  because  the 
deductions  were  drawn  from  infinite  series  without  first  ascertaining  that 
those  series  were  convergent.  This  defect  was  fatal,  and  to-day  La- 
grange's "  method  of  derivatives,"  as  his  method  was.  called,  has  been 
generally  abandoned  even  in  France. 

At  the  beginning  of  this  century  the  avidity  with  which  the  results 
of  modern  analysis  were  sought  began  so  far  to  subside  as  to  allow 
mathematicians  to  examine,  and  discuss  the  grounds  on  which  the  sev- 
eral principles  were  established.  The  doctrine  of  infinite  series  re- 
ceived its  due  share  of  attention.  In  building  up  a  tenable  theory  of 
infinite  series,  the  same  course  became  necessary  which  was  followed 
some  years  ago  in  the  erection  of  the  Washington  monument  in  the 
District  of  Columbia  j  after  the  work  had  proceeded  to  a  certain  height, 
the  old  foundation  was  found  to  be  insecure  j  it  had  to  be  removed  and 
to  be  replaced  by  another  which  was  broader  and  deeper.  The  engineer 
to  whom  more,  perhaps,  than  any  other  we  are  indebted  for  the  laying 
of  a  new  and  firm  foundation  to  infinite  series  and  to  analysis  in  general, 
is  Cauchy.  In  the  following  few  but  pregnant  sentences,  taken  from  his 
Cours  d^ Analyse  (Paris,  1821,  p.  2),  he  states  the  object  he  has  striven 
to  attain :  "  As  far  as  methods  are  concerned,  I  have  endeavored  to 
give  them  all  the  rigor  required  in  geometry,  and  never  to  have 
recourse  to  the  reasons  drawn  from  the  generalization  of  algebra. 
Reasons  of  that  kind,  although  they  are  very  generally  accepted,  espe- 


HISTOEICAL   ESSAYS.  365 

cially  in  passing  from  converging  series  to  diverging  series  and  from 
real  quantities  to  imaginary  quantities,  can  be  considered,  it  seems  to 
me,  only  as  inductions,  fit  to  give  a  glimpse  of  the  truth,  but  which 
agree  little  with  the  boasted  exactness  of  mathematical  science.  It  is 
furthermore  to  be  observed  that  they  tend  to  give  to  algebraic  formulae 
an  indefinite  extent,  while  in  reality  most  of  these  formulae  remain  true 
only  under  certain  conditions  and  for  certain  values  of  the  quantities 
which  they  contain."  These  weighty  words  of  Cauchy  became  the  parole 
of  a  new  scientific  party.  Oauchy  himself  was  eminently  successful  in 
his  work.    To  him  we  owe  the  first  correct  proof  of  Taylor's  theorem. 

He  took  very  strong  and  positive  grounds  against  the  use  of  diver- 
gent series.  All  series  that  were  not  convergent,  he  pronounced  falla- 
cious. Taylor's  theorem  he  considered  as  being  wrong  whenever  the 
series  became  divergent.  In  his  Cours  W Analyse  no  place  was  given  to 
those  troublesome  divergent  infinite  series  that  had  previously  been  the 
cause  of  so  much  vagueness,  uncertainty,  and  even  of  error. 

But  Cauchy  was  not  alone  in  this  protest  against  the  unrestricted 
use  of  the  time-honored  methods  of  analysis.  A  youthful  mathematician 
from  northern  Europe,  a  worthy  descendant  of  mighty  Thor,  sided  with 
the  French  mathematician  in  the  contest.  This  new  combatant  was  the 
youthful  Abel,  who,  though  he  died  at  the  premature  age  of  twenty- 
seven,  left  behind  him  an  imperishable  name.  As  in  the  times  of  myth 
and  fable,  Thor,  the  thunderer,  hurled  his  huge  hammer  against  the 
mountain  giants,  so-  Abel,  with  his  massive  intellectual  hammer,  dealt 
powerful  blows  against  some  of  the  mathematical  methods  of  his  time. 
Notice  an  extract  from  a  letter  written  by  him  in  1826,  which  expresses 
the  convictions  to  which  his  profound  studies  had  led  him.    Says  Abel  :* 

"Divergent  series  are  in  general  very  mischievous  affairs,  and  it  is 
shameful  that  any  one  should  have  founded  a  demonstration  upon  them. 
You  can  demonstrate  anything  you  please  by  employing  them,  and  it  is 
they  who  have  caused  so  much  misfortune,  and  given  birth  to  so  many 
paradoxes.    Can  anything  be  more  horrible  than  to  declare  that 

0=1— 2''+3"— i^+S"—  etc., 

when  w  is  a  whole  positive  number  ?  At  last  my  eyes  have  been  opened 
in  a  most  striking  manner,  for,  with  the  exception  of  the  simplest  cases, 
as  for  example  the  geometric  series,  there  can  scarcely  be  found  in  the 
whole  of  mathematics  a  single  infinite  series  whose  sum  has  been  rigor- 
ously determined ;  that  is  to  say,  the  most  important  part  of  mathe- 
matics is  without  foundation.  The  greater  part  of  the  results  are  correct, 
that  is  true,  but  that  is  a  most  extraordinary  circumstance.  I. am  en- 
gaged in  discovering  the  reason  of  this — a  most  interesting  problem. 
I  do  not  think  that  you  could  propose  to  me  more  than  a  very  small 
number  of  problems  or  theorems  containing  infinite  series,  without  my 
being  able  to  make  well-founded  objections  to  their  demonstration.    Do 

*  (Euvres  CompUtef  deN.  H.  Aiel,  Tome  I,  Christiania,  1839,  p.  264. 


366  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

SO,  and  I  will  answer  you.  Not  even  the  binomial  theorem  has  yet  been 
rigorously  demonstrated.    I  have  found  that 

/i.      \m      1    ,  .wi(w— 1)21     4. 

{l-{-  x)"'  =  l-{-mx-\-  — ^-y^ — -  a?  +  etc. 

for  all  values  of  x  which  are  less  than  1.  When  a?  =  4- 1,  the  same 
formula  holds,  but  only  provided  that  m  is  >  —  1 ;  and  when  a?  =  —  1, 
the  formula  only  holds  for  positive  values  of  m.  For  all  other  values 
of  m  the  series  \-\-mx-\-  etc.,  is  divergent.  Taylor's  theorem,  the  foun- 
dation of  the  whole  infinitesimal  calculus,  has  no  better  foundation.  I 
have  only  found  one  single  rigorous  demonstration  of  it,  and  that  is  the 
one  given  by  M.  Cauchy  in  his  Abstract  of  Lectures  upon  the  Infinitesi- 
mal Calculus,  where  he  has  demonstrated  that  we  have 

ij  (a;  +  CO  )  =i)  (a?)  +  ccj?'  {x)  +  ^  i?"  {x)  -f  etc., 

as  long  as  the  series  is  convergent ;  but  it  is  usually  employed  without 
ceremony  in  all  cases.    *    *    * 

"  The  theory  of  infinite  series  in  general  rests  upon  a  very  bad  foun- 
dation. All  operations  are  applied  to  them  as  if  they  were  finite ;  but 
is  this  permissible  ?  I  think  not.*  Where  is  it  demonstrated  that  the 
differential  of  an  infinite  series  is  found  by  taking  the  differential  of 
each  term  ?  Nothing  is  easier  than  to  give  examples  where  this  rule  is 
not  correct.  *  *  *  The  same  remark  holds  for  the  multiplication 
and  division  of  infinite  series.  I  have  begun  to  examine  the  most  im- 
portant rules  which  are  (at  present)  esteemed  to  hold  good  in  this  re- 
spect, and  to  show  in  what  cases  they  are  correct  and  in  what  not  so. 
This  work  proceeds  tolerably  well  and  interests  me  infinitely." 

Such  is  the  unequivocal  language  of  Abel.  His  early  death  prevented 
him  from  carrying  all  his  plans  into  execution.  To  him  we  are  indebted 
for  the  first  rigorous  proof  of  the  binomial  theorem.t 

The  views  on  infinite  series  held  by  Cauchy  and  Abel  met  with  hearty 
acceptance  by  leading  mathematicians  on  the  continent.  Thus,  Poisson 
expressed  his  views  in  the  following  language :  "  It  is  taught  in  the  ele- 
ments that  a  divergent  series  can  not  serve  to  calculate  the  approximate 
value  of  the  function  from  which  it  results  by  development,  but  some- 
times it  has  apparently  been  thought  that  such  a  series  can  be  used  in 
analytical  calculations  instead  of  the  function ;  and  although  this  error 
is  far  from  being  general  among  geometers,  nevertheless  it  is  not  useless 
to  point  it  out,  for  the  results  which  are  obtained  by  means  of  divergent 
series  are  always  uncertain  and  most  of  the  time  inexact." 

The  conditions  for  convergency  and  divergency  of  different  series  be- 

*  Dirichlet  first  pointed  out  that  the  most  elementary  algebraic  rule,  according  to 
which  every  sum  is  independent  of  the  arrangement  and  grouping  of  the  terms  to  be 
added,  does  not  necessarily  hold  true  in  infinite  series. 

t  (Euvres  Confutes  de  N,  M.  Abel,  Tome  Jj  Christiania,  1839,  p.  66. 


HISTORICAL   ESSAYS.  367 

gan  to  be  carefully  investigated.  No  universal  criterion  for  determin- 
ing whether  a  given  series  is  convergent  or  divergent  was  then  known  j 
nor  do  we  possess  such  a  one  even  to-day. 

A  question  naturally  arising  at  this  point  of  our  inquiry  is,  whether 
the  views  of  Cauchy  and  Abel  and  their  co-workers  met  at  once  with 
general  acceptance  or  not.  As  might  almost  be  expected,  they  did  not, 
but  encountered  firm  opposition.  The  old  combinatorial  school  in  Ger- 
many would  not  surrender  their  orthodox  views  without  a  struggle. 
They  obstinately  defended  every  doctrine  of  their  mathematical  creed. 
Even  such  a  man  as  Dr.  Martin  Ohm,  who  was  really  an  enemy  of  the 
combinatorial  school,  and  whose  achievements  in  mathematics  and 
physics  place  him  among  the  coryphaei  of  science,  was  not  willing  to 
join  Cauchy  and  Abel  in  calling  divergent  series  fallacious.  In  an  essay 
written  by  Ohm,  entitled.  The  Spirit  of  Mathematical  Analysis,*  he 
admits  that  the  great  mathematicians  of  his  day,  as  Gauss,  Diriohlet, 
Jacobi,  Bessel,  Cauchy,  do  not  employ  demonstrations  conducted  with 
divergent  series,  while  Poisson  speaks  decidedly  against  them.  "But," 
says  Ohm,  "  that  the  series  which  are  used  and  from  which  deductions 
are  drawn  ought  to  be  always  and  necessarily  convergent  is  a  circum- 
stance of  which  the  author  of  this  essay  has  not  been  able  at  all  to  con- 
vince himself ;  on  the  con  trary,  it  is  his  opinion  that  series,  as  long  as 
they  are  general,  so  that  we  can  not  speak  of  their  convergency  or  diver- 
gency, must  always,  when  properly  treated,  necessarily  and  uncondi- 
tionally produce  correct  results."  By  a  general  series  Ohm  means  one  in 
which  the  letters  represent  neither  magnitudes  nor  numbers,  but  are  con- 
sidered as  perfectly  insignificant  {inhaltlos).  Whenever  the  letters  are 
made  to  represent  magnitudes  or  numbers,  then  the  series  is  no  longer 
a  general  series,  but  is  a  "  numeric  "  series,  and  in  that  case  Ohm  admits 
that  an  equality  can  exist  between  the  function  and  its  series  only  when 
the  series  is  convergent.  It  is  very  diflSicultto  see  exactly  what  meaning 
shall  be  given  to  letters  upon  which  algebraic  operations  are  to  be  per- 
formed, when  the  letters  represent  neither  magnitudes  nor  numbers. 
Nor  is  it  easy  to  see  in  what  way  formulae  involving  these  empty,  mean- 
ingless letters — these  "  ghosts  of  departed  quantities  " — can  furnish 
rigorous  methods  in  mathematical  analysis.  In  fact,  this  theory  of 
general  series  containing  insignificant  letters  is  one  of  the  last  shifts  to 
which  the  opponents  of  the  new  school  resorted  5  one  of  the  last  sub- 
terfuges before  giving  up  a  contest  which  had  become  entirely  hopeless. 

If  we  pass  from  Germany  to  England  we  meet  there  with  another 
mathematician  who  championed  the  old  cause.  I  refer  to  George  Pea- 
cock, who  is  well  known  to  mathematicians  for  his  Algebra  and  his  Ee- 
port,  made  in  1833  to  the  British  Association,  On  the  Eecent  Progress 
and  Present  State  of  Certain  Branches  of  Analysis. 

Peacock  states  his  views  with  more  clearness  than  Ohm  had  stated 
his.    He  bases  his  argument  on  what  he  calls  the  "  principle  of  the  per- 

*The  Spirit  of  Mathematical  Analysis  and  its  Relation  to  a  Logical  System,  by  Dr. 
Martin  Ohm;  translated  by  Alexander  John  Ellis,  London,  1843. 


368  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

manence  of  equivalent  forms,"  wliicli  he  considers  to  be  tlie  real  foun- 
dation of  all  rules  of  symbolic  algebra.  According  to  this  principle,  all 
the  rules  and  operations  of  arithmetic  which  have  been  established  by 
numerical  considerations  are  adopted  without  reference  to  relative  mag- 
nitude ;  the  symbols  of  algebra  are  taken  to  be  perfectly  general  and 
unlimited  in  value,  and  the  operations  to  which  they  are  subject  are 
equally  general.  To  illustrate :  In  arithmetic  we  can  subtract  a  smaller 
number  from  a  larger,  but  we  cannot  subtract  a  larger  from  a  smaller ; 
that  is  to  say,  we  can  subtract  3  from  5,  but  not  5  from  3.  In  algebra, 
on  the  other  hand,  no  limitation  whatever  is  placed  upon  the  relative 
values  of  minuend  and  subtrahend ;  there  we  can  subtract  5  from  3 
and  give  the  answer  a  rational  interpretation.  By  the  principle  of  the 
permanence  of  equivalent  forms  every  result  obtained  from  mathemat- 
ical operations  must  always  be  a  correct  result,  no  matter  what  the 
relative  values  of  the  quantities  be  upon  which  the  operations  are  per- 
formed. Peacock  applies  this  principle  to  the  subject  of  infinite  series. 
He  says  (p.  205,  Eeport  for  1833)  that  "  the  series 

(1  +  a;) »  =  1"  (^1  +  wa;  +    ^  ^^  ~  ^^   x"  -\-  etc.^ 

indefinitely  continued,  in  which  w  is  a  particular  value  (a  whole  number), 
though  general  in  form,  must  be  true  also,  in  virtue  of  the  principle  of 
the  permanence  of  equivalent  forms,  when  n  is  general  in  value  as  well 
as  in  form."  Instead  of  being  always  a  positive  whole  number,  the  ex- 
ponent n  may,  therefore,  be  negative  or  fractional,  and  the  above  for- 
mula still  holds  true. 

Now,  the  principle  of  the  permanence  of  equivalent  forms  laid  down 
by  Peacock  is  not  self-evident,  nor  did  it  become  known  by  intuition ; 
on  the  contrary,  it  is  merely  an  induction,  and  can,  therefore,  hardly  be 
taken  as  a  reliable  basis  upon  which  to  settle  a  disputed  question ;  for 
this  very  question  may  be  one  in  which  this  law  established  by  mere 
induction  might  fail.  But  even  granting  the  principle  of  the  perma- 
nence of  equivalent  forms  to  be  generally  applicable,  does  it  really  fol- 
low from  it  that  infinite  series  are  true,  whether  they  be  convergent  or 
divergent  ?  In  order  to  discuss  this  point  let  us  examine  a  series  re- 
sulting from  the  division  of  the  numerator  of  an  algebraical  fraction  by 

its  denominator,  such  as . 

1— a 

From  arithmetic  we  get  the  simple  but  general  statement  that  the 
numerator  of  a  fraction  divided  by  its  denominator  is  equal  to  the  quo- 
tient plus  the  remainder  (if  there  be  any  remainder).  By  the  principle 
of  the  permanence  of  equivalent  forms  this  must  be  true  of  fractions 
involving  any  quantities  whatever.     Now,  if  we  divide  1  by  1  —  « 

we  get  1  -f  «  4-  a^  +  a^  -f  ^ .    We  oberve  there  is  a  remainder, 

a* 
:j .    If  we  carry  the  division  further  there  is  still  a  remainder.    No 


HISTORICAL   ESSAYS.  369 

matter  how  far  the  division  proceeds  it  will  not  end,  and  a  remainder 

will  still  exist.    We  may  express  tliis  fact  by  writing  :, =  1  +  a  + 

1  —  a 

0?  -\- a"  +  ..  _    .    Now,  if  a  has  a  value  less  than  unity  the 

remainder  approaches  zero,  and  we  may  therefore  write  t —  =  1  + 

a  ^  ^2  _j_  qIq^^  d^  infinitum.  This  infinite  series  is  correct  whenever 
a  <  1.  But,  according  to  Peacock,  it  would  follow  from  the  principle  of 
the  permanence  of  equivalent  forms  that,  if  this  series  is  correct  for 
a  <;1,  it  must  be  true  for  all  values  of  a.  Hence  the  series  is  true  when 
a  >  1,  in  which  case  the  series  is  divergent.  Now,  this  conclusion 
appears  to  be  inadmissible,  because  Peacock  does  not  examine  the 
remainder.  When  a  <  1,  the  remainder  approaches  zero,  and  can  there- 
fore be  neglected ;  but  if  a  >  1,  then  we  shall  find  that  the  remainder 
does  not  approach  zero,  and  therefore  cannot  be  neglected. 

To  neglect  it  would  be  to  violate  the  principle  of  the  permanence  of 
equivalent  forms.  This  principle  demands  that  whenever  there  is  a 
remainder  it  shall  always  be  considered  and  expressed,  no  matter  how 
far  the  division  be  continued.  If  in  the  above  series  we  take  a  =  2 
and  neglect  the  remainder,  then  we  get 

-1  =  1  +  2  + 22 +  23+ ad  infinitum, 

which  is  an  absurdity.  But  if  the  remainder  be  taken  into  account, 
then  we  have 

-  1  =  1  +  2  +  2«  +  23  +  .  .  .  2"  + -3^. 

This  equation  is  always  true,  no  matter  how  great  n  may  be ;  that  is  to 
say,  no  matter  how  far  the  division  be  continued.  From  similar  con- 
siderations in  other  series  it  would  appear  that  divergent  series  are 
false  and  absurd,  except  when  written  with  the  remainder. 

And  yet  not  only  Peacock,  but  even  De  Morgan  was  not  willing  to 
reject  divergent  series.  Though  De  Morgan  criticised  the  new  school 
for  the  unconditional  rejection  of  divergent  series,  he  cannot  be  pro- 
nounced an  enthusiastic  supporter  of  the  old  school.  In  an  article  in 
the  Transactions  of  the  Cambridge  Philosophical  Society,  Volume  VIII, 
Part  I,  he  says:  "I  do  not  pretend  to  have  that  confidence  in  series 
which,  to  judge  from  elementary  writers  on  algebra,  is  common  among 
mathematicians,  not  even  convergent  series."  His  views  on  this  sub- 
ject will  be  more  fully  elucidated  by  the  following  quotation  from  his 
article  on  "Series"  in  the' Penny  Cyclopedia:  "A  divergent  series  is, 
arithmetically  speaking,  infinite;  that  is,  the  quantity  acquired  by 
summing  its  terms  may  be  made  greater  than  any  quantity  agreed  on 
at  the  beginning  of  a  process.  *  *  *  Nevertheless,  as  every  alge- 
braist knows,  such  series  are  frequently  used  as  the  representatives  of 
881— No.  3 24 


370  TEACHING   AND    HISTORY    OF   MATHEMATICS. 

finite  quantities.  It  was  usual  to  admit  such  series  without  hesitation ; 
but  of  late  years  many  of  the  continental  mathematicians  have  declared 
against  divergent  series  altogether,  and  have  asserted  instances  in 
which  the  use  of  them  leads  to  false  results.  Those  of  a  contrary  opin- 
ion have  replied  to  the  instances,  and  have  argued  from  general  prin- 
ciples in  favor  of  retaining  divergent  series.  Our  own  opinion  is  that 
the  instances  have  arisen  from  a  misunderstanding  or  misuse  of  the 
series  employed,  though  sufficient  to  show  that  divergent  series  should 
be  very  carefully  handled ;  but  that,  on  the  other  hand,  no  perfectly 
general  and  indisputable  right  to  the  use  of  these  series  has  been  es- 
tablished a  priori.  They  always  lead  to  true  results  when  properly 
used,  but  no  demonstration  has  been  given  that  they  must  always  do 
so." 

About  the  time  when  Peacock  made  his  report  to  the  British  Asso- 
ciation, Cauchy  was  developing  new  and  valuable  results  on  the  subject 
of  infinite  series.  With  the  aid  of  the  integral  calculus  he  was  conduct- 
ing a  careful  investigation  of  the  conditions  which  must  be  fulfilled  in 
order  that  a  function  be  capable  of  being  developed  into  a  convergent 
infinite  series.  He  found  that  four  conditions  must  be  satisfied :  (1) 
The  function  must  admit  of  a  derivative.  (2)  The  function  must  be 
uniform^  that  is,  for  any  particular  value  for  x  the  function  must  have 
only  one  value.  (3)  The  function  must  be  finite.  (4)  The  function  must 
be  continuous,  that  is,  it  must  change  gradually  as  the  variable  passes 
from  one  value  to  another.  These  results  greatly  strengthened  the  posi- 
tion held  by  the  new  school,  and  notwithstanding  the  adroit  arguments 
brought  forth  by  various  mathematicians  of  the  old  school  in  favor  of  di- 
vergent series,  the  leading  mathematicians  of  to-day  have  rejected  the 
old  views  and  adopted  those  of  Cauchy  and  Abel.  In  the  theory  of  func- 
tions, a  branch  of  mathematics  which  is  now  assuming  enormous  propor- 
tions, the  convergency  of  all  series  employed  is  carefully  and  scrupulously 
tested.  In  late  years  more  reliable  criteria  have  been  invented  for 
determining  the  convergency.  Standard  treatises  on  the  subject  devote 
the  larger  part  of  their  space  to  the  consideration  of  convergency. 
Whenever  a  series  is  divergent,  then  either  the  remainder  is  inserted 
or  the  series  is  unceremoniously  rejected.  Indeed,  divergent  series  are 
now  looked  upon  by  our  best  mathematicians  as  being  nothing  more 
than  exploded  chimeras. 

Having  briefly  traced  the  history  of  infinite  series  in  Europe,  we  shall 
consider  the  views  on  this  subject  held  by  American  writers.  Previous 
to  the  beginning  of  this  century  the  text-books  on  algebra  used  in  this 
country  were  all  imported  from  abroad.  About  the  only  mathematical 
books  published  in  America  before  1800  were  arithmetics  and  some  few 
books  on  surveying.  The  earliest  imported  algebras  came  from  Great 
Britain.  The  most  important  of  them  were  the  algebras  of  Maclaurin, 
Saundersou,  Charles  Hutton,  John  Bonnycastle,  and  Thomas  Simpson. 
These  writers  belonged  to  what  we  have  called  the  old  school.    As 


HISTORICAL   ESSAYS.  371 

miglit  be  expected  the  subject  of  series  was  handled  by  them  with  the 
same  looseness  and  recklessness  as  by  the  older  school  of  mathemati- 
cians on  the  continent.  Thus,  in  Hutton's  Mathematics,  which  was  a 
standard  work  in  its  day,  considerable  attention  was  paid  to  series,  but 
the  terms  "convergent"  and  "divergent"  were  not  even  mentioned. 
The  earliest  American  compiler  of  a  course  of  mathematics  for  colleges 
was  Samuel  Webber.  In  1801  he  published  his  "  Mathematics."  The 
alge^braical  part  was  necessarily  elementary  in  character,  and  of  course 
contained  no  formal  criteria  for  convergency.  Whatever  defects  Web- 
ber's Algebra  may  have,  it  has  also  its  merits.  It  is  pleasing  to  observe 
that  as  far  as  the  author  had  entered  upon  the  subject  of  infinite  series 
he  was  on  the  right  track.  Speaking  of  a  certain  divergent  series  he 
says  that  *'  it  is  false,  and  the  further  it  is  continued  the  further  it  will 
diverge  from  the  truth"  (p.  291).  This  language  possesses  the  true 
ring ;  it  is  free  from  the  discords  of  error,  and  we  regret  that  American 
writers  of  later  date  have  not  imitated  it. 

In  1814,  thirteen  years  after  the  publication  of  Webber's  Mathemat- 
ics, appeared  the  algebra  of  Jeremiah  Day,  of  Yale  College.  All  things 
considered,  Day's  Algebra  is  superior  to  Webber's,  but  on  the  particu- 
lar subject  of  series  it  can  hardly  be  said  to  excel.  President  Day  points 
out,  to  be  sure,  that  a  certain  series  must  converge  in  order  to  come 
nearer  and  nearer  to  the  exact  value  of  the  fraction  from  which  the 
series  was  derived,  but  he  does  not  even  hint  at  the  insecurity  or  ab- 
surdity of  divergent  series.  He  gives  no  demonstration  of  the  binomial 
theorem,  but  speaks  of  it  as  being  universally  true. 

Four  years  after  the  publication  of  Day's  Algebra,  John  Farrar,  pro- 
fessor of  mathematics  at  Harvard,  published  An  Introduction  to  the 
Elements  of  Algebra,  »  *  *  Selected  from  the  Algebra  of  Euler.  On 
the  continent  of  Europe  Euler's  writings  were  at  that  time  justly  con- 
sidered as  the  most  profound,  and  as  affording  the  finest  models  of 
analysis.  Yet  his  writings  were  not  faultless.  His  views  on  series  were 
those  of  the  old  school.  The  discussion  of  series  as  given  in  Farrar's 
Euler  demands  our  attention,  because  subsequent  American  writers 
were  doubtless  greatly  influenced  by  it.    On  page  76  of  this  book,  the 

fraction  :; is  resolved  by  division  into  an  infinite  series.    The  follow- 

1—a 

ing  comments  upon  it  are  then  made :  "  There  are  sufficient  grounds  to 

maintain  that  the  value  of  this  infinite  series  is  the  same  as  that  of  the 

fraction .  What  we  have  said  may  at  first  seem  surprising,  but  the 

consideration  of  some  particular  cases  will  make  it  easily  understood. 
*    *    *    If  we  suppose  a=2,  our  series  becomes  =l-|-24-4-f8-f  16+32-f , 

64,  etc.,  to  infinity,  and  its  value  must  be  - — -  that  is  to  say,  — -  =  —  1, 

1 — 2  ^1 

which  at  first  sight  will  appear  absurd.    But  it  must  be  remarked  that 

if  we  wish  to  stop  at  any  term  of  the  above  series  we  can  not  do  so 


372  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

without  joining  the  fraction  which  remains."  Now  this  last  sentence  is 
certainly  a  true  statement.  E'o  fault  can  be  found  with  it.  It  simply 
means  that  we  must  consider  the  remainder,  the  very  thing  which  the 
new  school  persistently  insists  upon.  But  the  next  statement  made  by 
the  author  is  objectionable.  Says  he:  "Were  we  to  continue  the  series 
without  intermission,  the  fraction  indeed  would  be  no  longer  consid- 
ered, but  then  the  series  would  still  go  on."  This  really  amounts  to 
saying  that  when  the  series  becomes  infinite,  the  remainder  shall'  not 
be  considered. 

Now,  if  the  remainder  is  not  taken  into  account,  then  we  can  say  in 
the  language  of  Webber  that  the  further  the  series  is  continued,  "the 
further  it  will  diverge  from  the  truth,"  hence  it  must  be  "false." 

In  addition  to  this  abridgment  of  Euler's  Algebra,  Professor  Farrar 
published  a  translation  from  the  French  of  Lacroix's  Algebra.  La- 
croix's  works  are  justly  celebrated  for  their  purity  and  simplicity  of 
style.  Though  more  cautious  in  his  statements  than  the  majority  of  ele- 
mentary writers,  he  must  still  be  classed  as  belonging  to  the  old  school. 
In  his  algebra  *  he  speaks  of  divergent  series  as  leading  to  consequences 
that  are  "  absurd."  The  binomial  theorem  is  proved  by  Lacroix  for  the 
case  when  the  exponent  is  a  positive  integer,  but  the  proof  for  the  other 
cases  is  omitted.  In  the  light  of  modern  mathematics  this  was  a  wise 
omission.  A  correct  proof  of  the  general  theorem  is  too  difficult  for  pu- 
pils beginning  algebra.  But  the  easier  proofs  are  incorrect.  Hence  it 
is  preferable  to  give  no  proof  at  all  than  give  a  wrong  one.  But  Pro- 
fessor Farrart  was  not  satisfied  with  this  omission.  In  his  translation 
he  adds  a  foot-note,  with  the  erroneous  statement  that  the  binomial  for- 
mula is  "  equally  applicable  to  cases  in  which  the  exponent  is  fractional 
and  negative,"  and  he  demonstrates  this  theorem  in  the  last  part  of  his 
Cambridge  Course  of  Mathematics  ("  On  the  Differential  and  Integral 
Calculus")  without,  of  course,  considering  the  question  of  convergency. 

Charles  Davies,  who  was  appointed  professor  of  mathematics  at  the 
United  States  Military  Academy  at  West  Point  in  1823,  published  iu 
1834  an  algebra  modeled  after  the  large  French  treatise  of  Bourdon. 
This  algebra  is  familiarly  known  as  "Davies'  Bourdon,"  and,  like  all 
other  books  of  Professor  Davies,  has  had  a  very  extensive  circulation 
in  all  parts  of  the  United  States.  However  excellent  this  treatise  may 
be  in  other  respects,  on  the  subject  of  infinite  series  and  the  treatment 
of  the  binomial  theorem  it  is  very  defective. 

From  what  has  been  said  it  will  be  seen  that  the  foreign  authors 
whom  our  American  writers  took  for  models  in  compiling  their  algebras 
belonged  to  the  old  school.  Our  early  American  writers  clung  faith- 
fully to  the  orthodox  opinions  of  this  school.  The  only  dissenting 
voice  came  from  Samuel  Webber,  and  it  was  so  feeble  that  it  escaped 


I 


*  Elements  of  Algebra,  by  S.  F.  Lacroix,  translated  by  John  Farrar.  Second  edi- 
tion, Cambridge,  N.  E.,  1825,  p.  241.  (This  second  American  edition  was  translated 
from  the  eleventh  edition,  printed  in  Paris  iu  1815.) 

tliid,  p.  152. 


HISTORICAL   ESSAYS.  373 

all  notice.  But  what,  you  may  ask,  were  the  views  held  by  later  Amer- 
ican mathematicians  ? 

In  answer  to  this  I  need  not  discuss  each  author  individually.  If  we 
except  a  few  very  recent  writers,  then  we  may  say  that  on  infinite  series 
the  sins  of  one  are  quite  generally  the  sins  of  all.  You  may  consult 
the  large  and  extensive  Treatise  on  Algebra,  of  Charles  Hackley,  or  the 
Elementary  Treatise  on  Algebra,  by  James  Eyan ;  the  Elementary  and 
Higher  Algebra,  by  Theodore  Strong  j  the  University  Algebra,  by  Ho- 
ratio N.  Robinson ;  the  Algebra  for  Colleges  and  Schools  and  Private 
Students,  by  Joseph  Eay ;  the  Elements  of  Algebra,  by  Major  D.  H. 
Hill ;  the  University  Algebra,  by  Edward  Olney ;  the  Binomial  Theorem 
and  Logarithms,  by  William  Ohauvenet ;  the  Treatise  on  Algebra,  by 
Elias  Loomis.  You  may  consult  these  and  many  others,  and  you  will 
find  that  they  are  all  swayed  more  or  less  by  the  orthodox  ideas  of  the 
old  school.  A  few  of  them  give  tests  for  convergency,  but  none  of  them 
treat  divergent  series  with  that  severity  which  these  mischievous  ex- 
pressions deserve.  If  divergent  series  are  false,  then  it  ought  to  be  so 
stated;  the  student  should  be  informed  of  the  fact  that  they  are  false. 
Judging  only  from  American  algebras,  we  might  almost  conclude  that 
Cauchy,  Abel,  Poisson,  Dirichlet  had  never  lived,  or  that  their  ideas 
had  been  long  since  expunged  from  the  creed  of  true  science. ,  Of  the 
algebras  which  the  writer  has  examined  a  few  of  very  recent  date  are 
the  only  ones  to  which  this  statement  is  not  applicable.  But  even 
these  give  demonstrations  of  the  binomial  theorem  which  are  deficient 
in  rigor.  The  writer  has  not  seen  a  single  proof  of  this  theorem  for 
negative  or  fractional  exponents  in  any  American  algebra  which  is 
not  open  to  well-founded  objections.  Our  writers  often  begin  the  gen- 
eral proof  with  an  equation  in  which  the  sign  =  expresses  always  a 
numerical  equality,  and,  finally,  arrive  at  an  equation  (the  generalized 
binomial  formula)  in  which  the  sign  =  does  not  express  a  numerical 
equality,  except  under  certain  limiting  conditions.  The  student  is  not 
informed  in  what  way  such  a  change  in  the  meaning  of  the  sign  =  has 
been  brought  about,  nor  is  he  told  by  what  process  of  logic  this  sud- 
den metamorphosis  is  permissible. 

It  may  be  argued  that  the  final  equation  expresses  a  formal  truth. 
But  is  this  formal  truth  anything  more  than  a  perforated  shell  from 
which  the  kernel  of  useful  truth  has  been  removed  ?  When  the  equa- 
tion expresses  merely  a  formal  truth,  can  it  be  used  for  numerical  cal- 
culations? 'So.  Can  the  series  be  employed  in  course  of  analytical  dem- 
onstrations in  place  of  the  function  ?  No,  for  it  leads  to  uncertainty, 
and  perhaps  even  to  error.    What  then  is  this  formal  truth  good  for  ? 

There  is  an  American  algebraist  who  says  that  the  formula, 

(1-f  a;)»=l+ wic-f  ^^fcii  a^-f  etc., 

"  is  at  once  true  when  n  is  positive  or  negative,  entire  or  fractional, 
real  or  imaginary,  rational  or  irrational."    Yet,  it  was  pointed  out  long 


374"  TEACHING  AND  HISTORY   OF  MATHEMATICS. 

ago  by  Abel  and  others  that  even  when  the  conditions  for  convergency 
are  satisfied  there  are  still  other  points  to  be  considered  before  we  are 
entitled  to  write  the  sign  of  equality  between  the  function  and  the 
series.  The  expression  (l+a?)"  has  in  general  a  multiplicity  of  different 
values.  In  fact,  the  only  case  in  which  it  has  a  single  value  is  when 
the  exponent  n  is  an  integer.  Whenever  w  is  a  rational  fraction,  the 
expression  has  more  than  one  value ;  whenever  n  is  irrational  or  im- 
aginary, the  expression  has  an  infinite  number  of  values. 

The  series  itself,  on  the  other  hand,  has  always  only  one  value.  Now, 
if  we  place  the  function  (1  -\-  x)^  equal  to  the  series,  then  the  question 
arises,  which  one  out  of  the  possibly  infinite  number  of  functional  values 
is  equal  to  the  one  value  of  the  series  ? 

A  process  in  which  American  books  are  deficient  in  rigor  is  the  mul- 
tiplication of  one  infinite  series  by  another.  Some  of  our  books  exhibit 
not  the  slightest  hesitation  in  multiplying  by  one  another  any  two  series 
whatever,  and  placing  their  product  equal  to  the  product  of  the  functions 
from  which  the  two  series  were  obtained.  The  same  confidence  is  placed 
in  the  process  of  multiplying  infinite  series  as  in  that  of  multiplying 
fiiiite  expressions.  But  as  a  matter  of  fact,  when  one  or  both  series  are 
divergent,  then  their  product  is  an  absurd  result.  It  is  therefore  neces- 
sary that  both  series  be  convergent.  But,  strange  to  say,  this  neces- 
sary condition  has  not  always  been  found  sufScient.  There  are  cases  in 
which  the  product  of  two  convergent  series  may  actually  be  a  divergent 
series.    For  instance,  Cauchy  has  shown  that  the  series 

1_J_  +  J;^_J_.    i 

is  convergent,  but  that  its  square 

^-7r+(vr-^¥)-(jT  +  vr')+ 

is  divergent.  The  investigation  of  this  difiSculty  has  led  to  the  proof 
that  only  the  so-called  "  absolutely  convergent"  series  can  be  multiplied 
into  each  other  without  liability  of  error.  Thus  while  our  elementary 
books  teach  that  all  infinite  series  can  be  multiplied  by  one  another, 
the  most  recent  and  most  advanced  treaties  on  the  subject  teach  that 
only  convergent  series  of  a  particular  kind  can  be  multiplied  into  each 
other  so  as  to  lead  to  trustworthy  results. 

If  we  had  time  we  could  go  on  and  examine  the  development  of  func- 
tions into  series  by  the  method  of  indeterminate  co-efficients,  as  taught 
in  our  elementary  books.  We  should  meet  with  several  points  which 
are  open  to  well-founded  objections.  But  we  can  not  enter  upon  this 
subject  now. 

The  writing  of  a  good  elementary  text-book  is  one  of  the  most  diffi- 
cult undertakings.  It  is  hardly  advisable  to  subject  to  rigorous  proof 
every  rule  and  every  process  which  ought  to  go  into  an  elementary 
text-book  on  algebra. 


HISTOEICAL   ESSAYS.  375 

Many  of  the  proofs  would  be  either  esceedingly  difficult  to  the  pupil 
or  entirely  beyond  his  comprehension.  In  consequence  of  this  the  prob- 
lem arises  to  decide  what  had  best  be  proved  and  what  might  best  be 
assumed  without  demonstration.  It  is  easy  to  see  how  the  opinions  of 
able  and  experienced  teachers' may  differ  in  making  this  choice.  But 
there  is  one  point  upon  which  there  should  be  no  difference  of  opinion. 
Whatever  is  placed  in  an  elementary  text-book  ought  to  be,  as  near 
as  we  can  make  it,  the  truth  and  nothing  but  the  truth.  If  a  subject  is 
so  difficult  that  it  can  not  be  stated  in  an  elementary  way  without  mis- 
stating  it,  then  it  had  best  be  left  out  altogether.  Whatever  reasoning 
would  be  fallacious  and  wrong  when  placed  in  an  advanced  treatise 
must  be  equally  fallacious  and  wrong  when  placed  in  an  elementary 
book.  If  divergent  series  are  unreliable,  absurd,  or  false  in  advanced 
articles  written  by  Oayley  and  Abel,  in  the  Cours  d' Analyse  by  Jordan 
•  and  by  Cauchy,  or  in  the  Calcul  Bifferentiel  by  Serret,  then  divergent 
series  must  be  equally  unreliable,  absurd,  or  false  in  the  elementary 
algebras  of  Loomis,  Davies,  or  Eobinson.  Now,  if  divergent  series  are 
actually  untrustworthy  and  fallacious  (and  the  leading  mathematicians 
of  to-day  consider  them  so),  would  it  not  be  best  to  make  a  statement 
to  that  effect  in  our  elementary  algebras  and  to  give  at  least  some  of  the 
simplest  criteria  for  determining  the  convergency.  If  a  correct  proof 
of  the  binomial  theorem  for  negative  and  fractional  exponents  is  too 
long  and  difficult  to  find  a  place  in  an  elementary  algebra,  why  should 
it  not  be  entirely  omitted  from  algebra,  and  inserted  afterward  in  the 
differential  and  integral  calculus  ?  There  it  can  be  deduced  at  once  as 
an  immediate  consequence  of  Taylor's  theorem.  But  in  that  case  we 
must  be  sure  that  our  calculus  gives  a  correct  proof  of  Taylor's  theorem. 

Unfortunately,  many  of  our  American  works  give  what  may  bo  called 
the  old  proof  of  this  theorem,  which  proof  is  pronounced  unsatisfactory 
by  all  standard  writers  on  the  calculus.  De  Morgan  does  not  consider 
it  a  demonstration  at  all,  but  treats  this  old  process  as  "  nothing  more 
than  rendering  it  highly  probable  that 

^{a  +  A)  and  ^{a)  +  ^'{a)  h  +  ^"{a)  ^  +  etc., 

have  relations  which  are  worth  inquiring  into."  Todhunter  likewise  ob- 
jects to  the  old  proof,  and  especially  to  "  the  use  of  an  infinite  series 
without  ascertaining  that  it  is  convergent." 

We  regret  to  say  that  many  of  our  American  books  on.  calculus  are 
just  as  reckless  and  unscrupulous  in  the  treatment  of  infinite  series  as 
our  algebras  are.  But  this  assertion  can  not  be  applied  sweepingly  to 
all  our  works  on  this  subject.  Take  some  of  the  more  recent  publica- 
tions, as,  for  instance,  Byerly's  Calculus.  On  page  118  of  Byerly's  Dif- 
ferential calculus  the  following  statement  is  made  and  emphasized  by 
italics.  "J*  is  very  unsafe  to  make  use  of  divergent  series  or  to  base  any 
reasoning  upon  tJiemJ^  This  doctrine  contradicts  the  doctrine  taught  in 
our  algebras.    If  Byerly's  Calculus  is  correct,  then  our  algebras  must 


376  TEACHING   AND   HISTOEY   OF   MATHEMATICS. 

be  wrong.  Imagine  the  confusion  which  will  arise  in  the  mind  of  the 
student.  While  he  is  studying  algebra  he  learns  that  the  binomial 
theorem  is  universally  true.  When  Byerly's  Calculus  is  placed  in  his 
hands,  he  discovers  that  this  same  theorem  is  not  always  true,  but 
holds  good  only  when  certain  conditions  are  satisfied.  The  thoughtful 
student  will  become  disgusted  at  such  glaring  contradictions  in  the  pre- 
sentation and  explanation  of  a  science  which,  in  the  hands  of  a  careful 
mathematician,  can  be  made  to  be  the  most  accurate  and  consistent  of 
all  sciences.  In  closing,  we  give  the  following  summary  of  the  views 
presented  in  this  paper : 

1.  In  calculating  with  or  reasoning  by  means  of  infinite  series,  the 
question  of  convergency  should  always  be  considered.  If  a  series  is 
divergent,  then  the  sign  of  equality  should  not  be  placed  between  that 
series  and  the  function  from  which  it  was  developed.  If  the  sign  of 
equality  be  used  in  that  way,  then  it  expresses  an  absurdity,  which  is  no 
less  an  absurdity  when  found  in  an  elementary  text-book  than  when 
found  in  a  more  advanced  treatise. 

2.  Those  parts  of  the  subject  which  are  too  difficult  for  correct  treat- 
ment in  algebra,  may  be  assumed  temporarily  without  demonstration, 
and  may  afterward  be  proved  in  the  differential  and  integral  calculus. 
This  suggestion  applies  particularly  to  the  binomial  formula  for  all 
cases  in  which  its  exponent  is  not  a  positive  integer. 


ON  PARALLEL  LINUS  AND  ALLIED  SUBJECTS. 

There  are  few  subjects  in  mathematics  which  have  been  discussed  to 
greater  extent  than  that  of  parallel  lines.  The  various  attempts  at  im- 
proving the  theory  of  this  subject  may  be  classified  under  four  heads  : 
I.  In  which  a  new  definition  of  parallel  lines  is  suggested.  II.  In  which 
a  new  axiom,  different  from  Euclid's,  is  proposed.  III.  In  which  efforts 
have  been  made  to  deduce  the  theory  of  parallels  from  the  nature  of 
the  straight  line  and  plane  angle.  IV.  In  which,  during  the  present 
century,  the  whole  subject  of  geometrical  axioms  has  been  re-investi- 
gated and  searched  to  the  very  bottom,  and  in  which  the  novel  and 
startling  conclusion  has  been  reached  that  the  space  defined  by  Euclid's 
axioms  is  not  the  only  possible  non -contradictory  space.  This  gave 
birth  to  what  is  now  termed  non-Euclidian  geometry. 

It  is  our  intention  to  take  up  the  discussion  under  the  above  four 
heads,  with  a  view  of  presenting  the  ideas  advanced  by  American  math- 
ematicians or  given  in  text-books  used  in  this  country. 

I.- -NEW  DEFINITIONS. 

Euclid's  definition  of  parallel  lines  is  as  follows:  ^^ Parallel  straigM 
lines  are  such  as  are  in  the  same  plane,  and  which  heing  produced  ever  so  far 
both  ways  do  not  meef^    This  definition  has  been  retained  by  the  larger 


HISTORICAL  ESSAYS.  377 

number  of  American  writers,^  and  seems  indeed  the  most  desirable  one 
to  use  in  elementary  geometry. 

Parallel  lines  are  lines  everywhere  equally  distant.  This  definition  has 
been  adopted  by  Hutton,^  Webber,^  T.  Walker,*  A.  Schuyler,^  and 
probably  by  other  authors  whose  books  have  not  been  examined  by  the 
writer.  This  definition  has  never  been  popular  here  since  the  time  of 
Webber.  Chief  among  older  and  foreign  authors  who  used  it  are  Wolf, 
Diirer,  Boscovich,  T.  Simpson  (in  the  first  edition  of  his  Elements),  and 
Bonnycastle.  Olavius  assumed  that  a  line  which  is  everywhere  equi- 
distant from  a  straight  line  is  itself  straight.  This  axiom  or  postulate, 
which,  by  the  way,  does  not  hold  true  in  pseudo-spherical  space  (accord- 
ing to  the  ordinary  methods  of  measurement),  lies  hidden  in  disguise  in 
the  above  definition.  The  objections  to  that  definition  are  that  it  Is  an 
advanced  theorem,  rather  than  a  definition  5  that  it  involves  a  number  of 
considerations  of  great  subtlety ;  and  that  it  has  to  be  abandoned  as  a 
fundamental  definition  in  the  more  generalized  view  which  is  taken  of 
this  science  in  what  is  called  non-Euclidian  geometry.  To  be  rejected 
for  similar  reasons  is  the  following  definition. 

Two  lines  that  make  equal  angles  with  a  third  line,  all  being  in  the  same 
plane,  are  parallel.  This  is  given  by  H.  N".  Bobinson.^  It  was  used  in 
France  by  Yarignon  and  Bezout,  and  in  England  by  Cooley. 

Parallel  lines  are  straight  lines  which  have  the  same  direction.  This  defi- 
nition has  been  growing  in  favor  in  this  country.  The  reason  of  its  popu- 
larity lies  in  the  fact  that  it  appears  to  contribute  to  the  brevity  and  sim- 
plicity of  demonstrations.  Its  validity  will  be  considered  further  on. 
One  of  the  first,  perhaps  the  first,  to  use  it  in  this  country  was  James  Hay- 
ward,  teacher  of  mathematics  at  Harvard  College."'  It  was  used  by 
Benjamin  Peirce,^  N.  Tillinghast,^  Charles  W.  Hackley,^^  Davies  and 
Peck,"  Eli  T.  Tappan,'*  William  T.  Bradbury,!^  and  G.  A.  Wentworth.^* 

In  England  the  concept  of  direction  was  made  the  basis  of  a  work 

1  Of  the  books  examined  by  the  writer,  the  following  employ  this  definition :  Far- 
rar,  F.  H.  Smith,  and  Davies,  in  their  respective  editions  of  Legendrie ;  also,  Chau- 
venet,  Newcomb,  Venable,  Halsted,  Loomis,  Grund,  Olney,  Hassler,  Hunter,  Whitlock, 
and  Wentworth  (in  the  revised  edition  of  his  geometry,  1888). 

sHutton's  Mathematics,  edited  by  Robert  Adrain,  New  York,  1831,  Vol.  I.,  p.  275. 

3  Webber's  Mathematics,  Cambridge,  N.  E.,  1808,  p.  340. 

*  Walker's  Elements  of  Geometry,  Boston,  1831,  p.  30. 

5  Schuyler's  Elements  of  Geometry,  1876,  p.  33. 

6  Elements  of  Geometry,  Plane  and  Spherical  Trigonometry,  Cincinnati,  1852,  p.  11. 
''  Geometry,  Cambridge,  1829,  p.  7. 

8  Elementary  Treatise  on  Plane  and  Solid  Geometry,  Boston,  1837. 
"  Plane  Geometry  for  the  use  of  schools,  Concord  and  Boston,  1841. 
1°  Elementary  Course  of  Geometry  for  the  use  of  schools  and  colleges,  New  York, 
1847. 
"  Mathematical  Dictionary,  Article,  "  Parallel  Lines." 
"  Treatise  on  Plane  and  Solid  Geometry,  "  Ray's  Series,"  Cincinnati,  1864. 
13  Elementary  Geometry  and  Trigonometry,  Boston,  1872. 

"  Elements  of  Plane  and  Solid  Geometry,  Boston,  1878.    All  editions  of  this  mo3t 
popular  book,  except  the  revi$ed  edition  of  June,  1888,  contain  the  above  definition  of 
-parallels. 


378  TEACHING  AND  HISTORY   OF   MATHEMATICS. 

on  geometry  by  J.  M.  TVilson,  1868,  but  in  his  new  book  of  1878  the 
whole  theory  of  direction  is  ignored. 

II. — NEW  "AXIOM." 

Euclid  proves  in  his  Elements  (I,  27)  that  "  If  a  straight  line  falling 
on  two  other  straight  lines  make  the  alternate  angles  equal  to  one 
another,"the  two  straight  lines  shall  be  parallel  to  one  another,"  But 
before  any  other  step  can  be  made,  it  is  necessary  either  to  prove  or 
assume  that  in  every  other  case  the  two  lines  are  not  parallel. 

Being  unable  to  prove  this,  Euclid  assumed  it.  His  assumption  con- 
sists in  what  is  generally  called  the  twelfth,  by  some  the  eleventh 
"axiom :"  "  If  a  straight  line  meet  two  straight  lines  so  as  to  make  the 
two  interior  angles  on  the  same  side  of  it  taken  together  less  than  two 
right-angles,  these  straight  lines,  being  continually  produced,  shall  at 
length  meet  on  that  side  on  which  are  the  angles  which  are  less  than 
two  right-angles."  It  has  been  validly  urged  against  Euclid  that  this 
statement  is  far  from  being  axiomatic.  But  H.  Hankel*  has  shown 
that  Euclid  himself  placed  this  among  the  postulates  (where  it  more 
properly  belongs)  and  not  among  the  axioms.  The  mistake  of  calling 
it  an  axiom  was  due  to  later  editors.  Euclid  thus  placed  the  whole 
difficulty  of  parallel  lines  in  an  assumption. 

It  has  been  objected  that  this  assumption  is  not  sufficiently  simple 
and  obvious.  Accordingly,  Play  fair  proposed  the  following  "  axiom" : 
"  Two  straight  lines  which  intersect  one  another  can  not  be  both  par- 
allel to  the  same  straight  line."  This  is  merely  Euclid's  "  axiom"  in  a 
better  and  more  obvious  form.  It  has  been  adopted  by  the  best  Ameri- 
can works  on  geometry.t 

A  large  number  of  our  geometries  give  neither  Euclid's  nor  Playfair's 
"  axiom,"  but  pretend  to  prove  some  "  theorem  "  which  states,  in  sub- 
stance, what  is  equivalent  to  Euclid's  "  axiom."  This  leads  us  to  the 
next  heading. 

III. — "  PKOOFS." 

Since  neither  Euclid's  nor  Playfair's  "  axiom  "  is  axiomatic,  innumer- 
able attempts  have  been  made  io  j^rove  one  or  the  other.  Until  within 
twenty  years  it  was  believed  by  many  leading  mathematicians  that 
valid  proofs  could  be  deduced  from  reasonings  on  the  nature  of  the 
straight  line.  But  the  researches  which  led  to  the  development  of  non- 
Euclidian  geometry  have,  at  last,  made  it  clear  that  all  such  attempts 
must  necessarily  remain  fruitless. 

We  shall  call  attention  to  a  few  so-called  proofs  found  in  text-books 
used  in  this  country.    Huttonf  proves  the  "  theorem  "  that  "  when  a  line 

*  Vorlesungen  ilher  CompUxe  ZaMen  und  Hire  FunMionen,  p.  52. 

tThe  -writer  has  seen  it  in  the  geometries  of  Davies,  F.  H.  Smith,  Venable,  Loomis, 
Chauvenet,  Hunter,  Brooks,  Grund,  Newcomb,  Halsted,  and  Wentworth  (in  his  re- 
vised edition,  1885). 

\  Hutton's  Mathematics,  edited  by  Eobert  Adrain,  New  York,  1831,  Vol.  I,  p.  288. 


HISTORICAL  ESSAYS.  379 

intersects  two  parallel  lines,  it  makes  the  alternate  angles  equal  to  each 
other  "  (which  is  the  equivalent  of  Euclid's  ''  axiom  ") — in  the  following 
manner :  If  angles  AEF  and  EFD  are  not  equal, 
"  one  of  them  must  be  greater  than  the  other;  let  it 
be  EFD,  for  instance,  which  is  the  greater,  if  pos- 
sible, and  conceive  the  line  FB  to  be  drawn,  cut- 
ting off  the  part  or  angle  EFB  equal  to  the  angle 
AEF,  and  meeting  the  line  AB  in  the  ])oint  B.  Then, 
since  the  outward  angle  AEF,  of  the  triangle  BEF, 
is  greater  than  the  inward  opposite  angle  EFB  (th.  8) ;  and  since  these 
two  angles  also  are  equal  (by  the  construction),  it  follows  that  those 
angles  are  both  equal  and  unequal  at  the  same  time,  which  is  impossi- 
ble. Therefore,  the  angle  EFD  is  not  unequal  to  the  alternate  angle 
AEF ;  that  is,  they  are  equal  to  each  other."  The  error  of  this  proof 
lies  in  the  (implied)  statement  that  the  line  FB  must  always  intersect 
the  line  AB^  which  is,  virtually,  an  assumption  of  the  thing  to  be  proved. 
We  know  that  in  pseudo-spherical  geometry  one  of  the  angles  (say  EFD) 
is  greater  than  the  other,  and  that  the  line  FB  does  not  cut  AB. 

This  same  proof  is  given  in  Davies'  Elementary  Geometry,  p.  26.  At- 
tempts at  proving  the  "  parallel-axiom  "  were  made  also  by  Hassler,  by 
a  writer  (James  Wallace)  in  the  Southern  Review  (Vol.  1, 1828),  and  by 
A.  0.  Twining,  in  Silliman's  Journal  (1846,  pp.  47  and  89). 

Olney*  proves  Playfair's  axiom  in  this  way :  "  Let  AB  be  the  given 
line,  and  Q  the  given  point,  there  can  be  one  and  only  one  perpendic- 
ular through  Q  to  AB  (127).  Let  this  be  FE.  Now  through  (}  one  and 
only  one  perpendicular  can  be  drawn  to  FE.  Let  this  be  CD.  Then  is  CD 
parallel  to  AB  by  the  proposition  (just  proved  in  the  book),  and  it  is 
the  only  such  parallel,  since  it  is  the  only  perpendicular  to  FE  at  (7." 
The  fallacy  of  this  lies  in  the  assumption  that  every  line  in  the  plane, 
drawn  through  the  point  G  and  not  cutting  the  line  AB  must  neces- 
sarily be  perpendicular  to  EF. 

An  interesting  attempt  to  prove  Euclid's  axiom  is  given  anonymously 
in  Crelle's  Journal  (1834),  and  translated  and  published  by  W.  W. 
Johnson  in  the  Analyst  (Vol.  Ill,  1876,  p.  103).  According  to  De  Mor- 
gan, this  proof  is  due  to  Bertrand.  Professor  Johnson  says  that 
"  this  demonstration  seems  to  have  been  generally  overlooked  by  writ- 
ers of  geometrical  text-books,  though  apparently  exactly  what  was 
needed  to  put  the  theory  upon  a  perfectly  sound  basis."  The  error  in 
the  proof  seems  to  lie  in  the  statement  that,  if  lines  AB  and  CD  in  a 
plane  lie  on  the  same  side  of  the  line  AG  and  are  equally  inclined  to  it, 
then  the  infinite  space  BAGD  must  always  be  less  than  the  infinite 
space  BAE,  provided  only  that  angle  BAE  be  not  taken  less  than 
angle  BAG.  That  this  is  not  true  in  spherical  geometry,  is  seen  very 
readily ;  nor  is  it  true  in  pseudo-spherical  geometry. 

*  Treatise  on  Special  or  Elementary  Geometry,  University  edition,  New  York,  1872, 
p.  70.    In  later  editions  this  proof  is  omitted  and  Playfair's  axiom  assumed. 


380  TEACHING   AND   HISTOEY   OF   MATHEMATICS. 

Those  aathors  who  adopfc  the  idea  of  "direction,"  and  define  par- 
allels as  lines  having  the  same  direction,  dispose  of  the  whole  subject 
in  a  trice.  To  them  the  theory  of  parallels  gives  no  trouble.  The 
difficulties  of  the  subject  are  all  hidden  from  sight  by  the  notion  of  "  di- 
rection." The  following  is  the  proof  given  by  Hayward*  of  a  "the- 
orem "  which  says,  in  substance,  the  same  thing  as  the  "parallel-axiom." 
"  The  straight  line  has  the  same  direction  in  every  part.  An  angle  is 
the  inclination  of  one  straight  line  to  another  5  that  is,  the  inclination 
to  each  other  of  these  two  directions.  Two  parallel  straight  lines  have 
the  same  direction.  Therefore,  a  straight  line  (which  has  but  one  di- 
rection in  every  part),  meeting  two  straight  lines  which  have  but  one 
direction  in  all  their  parts,  must  have  the  same  inclination  to  both. 
That  is,  icken  a  straight  line  meets  two  parallel  straight  lines,  the  angles 
which  it  malces  icith  the  one  are  equal  to  those  which  it  mahes  with  the  other. 
Clearer  evidence  of  the  truth  of  this  proposition  can  not  be  desired." 
A  little  further  on  we  shall  consider  the  question,  Is  the  directional 
method  scientific  ? 

A  mathematician  whose  attempts  to  prove  the  "parallel-axiom"  were 
awarded  with  the  most  fruitful  results,  was  M.  Legendre.  In  the  earli- 
est editions  of  his  celebrated  Elements,  he  makes  a  direct  appeal  to  the 
senses.  In  the  seventh  edition  he  assumes  that  a  magnitude  increases 
without  limit  when  perpetual  increase  is  all  that  is  demonstrable.  But 
his  early  proofs  of  the  "  parallel-axiom"  did  not  satisfy  even  him,  and  he 
temporarily  returned  to  Euclid's  mode  of  treating  parallels.  Farrar's 
second  edition  of  Legendre,  brought  out  in  1825,  contains  this  last  pre- 
sentation of  the  subject.  Further  investigations  led  Legendre  to  the 
beautiful  result  that  the  theory  of  parallels  can  be  strictly  deduced,  if 
it  can  previously  be  shown  that  the  three  angles  of  a  triangle  are  equal 
to  two  right  angles.  In  Farrar's  Legendre  of  1831  and  1833  is  given 
Legendre's  attempt  to  prove  this  theorem  without  previously  assuming 
the  "  parallel-axiom."  The  attempted  proof  is  somewhat  long,  and  in- 
troduces sfn  infinite  series  of  triangles.  In  Volume  XII  of  the  Memoirs 
of  the  Institute  is  a  paper  by  Legendre,  containing  his  last  attempt  at 
a  solution  of  the  problem.  Assuming  space  to  be  infinite,  he  proved 
satisfactorily  that  it  is  impossible  for  the  sum  of  the  three  angles  of  a 
triangle  to  exceed  two  right  angles ;  and  that  if  there  be  any  triangle 
the  sum  of  whose  angles  is  two  right  angles,  then  the  same  must  be 
true  of  all  triangles.  But  in  the  next  step,  to  show  that  this  sum  can 
not  be  less  than  two  right  angles,  his  demonstration  failed. 

IV.— EECENT  RESULTS. 

Some  years  before  Legendre  completed  the  above  investigations,  Lo- 
batchewsky  of  Russia  adopted  the  bold  plan  of  constructing  a  geom- 
etry without  assuming  the  parallel-axiom.     He  succeeded  in  this,  and 

•  Elements  of  Geometry,  p.  ix. 


HISTORICAL   ESSAYS.  381 

it  opened  up  tlie  subject  of  non-Euclidian  geometry.  His  discoveries 
were  first  made  public  in  a  discourse  at  Kasan,  February  12,  1826.  We 
can  not  here  discuss  the  investigations  on  this  subject  that  were  made 
by  Lobatchewsky,  Gauss,  Bolyai,  Beltrami,  Eiemann,  Helmholtz,  Klein, 
and  our  own  Newcomb.* 

We  shall  only  state  that  the  possibility  of  constructing  geometries 
upon  different  assumptions  than  those  made  by  Euclid  has  become  evi- 
dent.* We  know  now  that,  assuming  space  to  be  of  uniform  curvature, 
there  are  really  three  sorts  of  geometries  possible — those  of  spherical 
space,  of  Euclid's  space,  and  of  pseudo-spherical  space.  Each  of  these 
is  consistent  in  itself.  These  three  do  not  contradict  each  other,  but 
form  rather  one  great  system  of  which  each  is  only  a  special  case. 

Much  light  has  been  thrown  by  the  above  generalizations  upon  the 
vexed  subject  of  geometric  "  axioms."  Do  our  more  recent  text-books 
profit  by  these  researches?  Some  of  them  do.  Take,  for  instance, 
Newcomb's  Elementary  Geometry.  On  page  14  we  read,  "  We  are  to 
think  of  the  geometric  figures  as  made  of  perfectly  stiff  lines  which  can 
be  picked  up  from  the  paper  and  moved  about  without  bending  or 
undergoing  any  change  of  form  or  magnitude."  This  statement  em- 
braces a  property  that  is  a  common  characteristic  of  all  three  geome- 
tries mentioned  above,  and  distinguishes  that  group  from  any  other 
which  might  be  conceived,  namely,  the  property  that  a  figure  can 
be  moved  about  without  undergoing  either  stretching  or  tearing.f 
"  Through  a  given  point  one  straight  line  can  be  drawn,  and  only  one, 
which  shall  be  parallel  to  a  giv«n  straight  line."  The  assumption, 
"  one  straight  line  can  be  drawn,"  divides  the  Euclidian  and  pseudo- 
spherical  geometries  from  the  spherical  geometry  ;  for  in  the  last  there 
are  no  (real)  straight  lines  that  are  parallel  to  each  other.  The  assump- 
tion contained  in  the  words  "and  only  one,"  separates  Euclidian  geom- 
etry from  the  pseudo-spherical.  In  the  latter  more  than  one  line  can 
be  drawn  through  the  same  point,  none  of  which  intersect  a  given 
straight  line.  The  assumptions  thus  made  completely  define*  the  geom- 
etry of  Euclid  from  the  other  two.  A  good  statement  of  the  assump- 
tions about  Euclid's  space  is  found  also  in  flalsted's  Geometry. 

This  may  be  a  convenient  place  to  inquire  into  the  scientific  value  of 
the  term  "direction"  as  a  fundamental  geometric  concept.  Professor 
Halsted  says,  in  the  preface  to  his  geometry  :  "  In  America  the  geome- 
tries most  in  vogue  at  present  are  vitiated  by  the  immediate  assumption 
and  misuse  of  that  subtle  term,  'direction;'  and  teachers  who  know 
something  of  the  non-Euclidian,  or  even  the  modern  synthetic  gometries, 
are  seeing  the  evils  of  this  superficial  '  directional '  method.    *     *    * 

*Foi'  a  bibliography  of  hyper-space  and  non-Euclidian  geometry,  see  a  paper  by 
George  Bruce  Halsted  in  the  American  Journal  of  Mathematics,  Vol.  I,  pp.  261-276, 
and  pp.  384,  3S3  ;  Vol.  II,  pp.  65-70. 

t  The  property  that  figures  can  be  moved  about  "  without  bending"  distinguishes 
the  geometry  on  an  ordinary  plane  or  on  a  sphere  from  that  on  a  surface  like  the  coae. 


382  TEACHING   AND    HISTORY    OP   MATHEMATICS. 

The  present  work,  composed  with  special  reference  to  use  in  teaching, 
yet  strives  to  present  the  elements  of  geometry  in  a  way  so  absolutely 
logical  and  compact,  that  they  may  be  ready  as  rock-foundation  for 
more  advanced  study."  We  quote  on  this  subject  also  from  a  prominent 
German  work  of  Dr.  Wilhelm  Killing.*  "  The  attempts  to  establish  a 
natural  basis  for  geometery  have,  thus  far,  not  been  accompanied  with 
desired  success.  The  reason  for  that  lies,  in  my  opinion,  in  this,  that 
even  as  geometry  has  been  compelled  to  abandon  the  concept  of  direc- 
tion {Begriff  der  Riclitung)  in  the  senserequired  by  the  parallel-axiom,  so 
it  will  not  be  able  to  hold  on  to  the  concept  of  distance  {Begriff  des  Ah- 
standes)  as  a  fundamental  concept,  and  must,  therefore,  pass  far  beyond 
the  non-Euclidian  forms  of  space  (Baumformen)  in  the  narrower  sense." 
Thus,  according  to  Dr.  Killing,  geometry  has  discarded  the  term  direc- 
tion as  a  fundamental  concept. 

There  are  several  objections  which  can  be  urged  against  the  term 
"  direction."  When  we  think  of  two  straight  lines  as  having  different 
directions,  we  imagine  ourselves  placed  on  the  point  of  intersection  and 
looking  along  one  of  the  lines,  then  the  other.  The  term  seems  clear 
as  long  as  we  apply  it  to  lines  which  intersect  each  other,  or  which 
coincide  with  one  another.  In  the  latter  case  we  say  that  the  two  lines 
have  the  same  direction.  But  we,  as  yet,  have  no  geometric  meaning 
of  the  phrase  "the  same  direction,"  except  when  used  of  lines  having 
a  common  ]3oint.  Simply  because  lines  which  intersect  each  other  have 
different  directions,  we  can  not  logically  conclude  that  lines  which  do 
not  meet  each  other  have  the  same  direction.  This  objection  was  urged 
by  De  Morgan  twenty  years  ago  in  his  review  of  J.  M.  Wilson's  geome- 
try.f  Says  he,  "  There  is  a  covert  notion  of  direction,  which,  though 
only  defined  with  reference  to  lines  which  meet,  is  straightway  trans- 
ferred to  lines  which  do  not.  According  to  the  definition,  direction  is 
a  relation  of  lines  which  do  meet,  and  yet  lines  which  have  the  same 
direction  can  be  lines  which  never  meet.  *  *  *  How  do  you  know, 
we  ask,  that  lines  which  have  the  same  direction  never  meet  ?  Answer — 
lines  which  meet  have  ^(^erewit  directions.  We  know  they  have ;  but  how 
do  we  know  that,  under  the  definition  given,  the  relation  called  direction 
has  any  application  at  all  to  lines  which  never  meet  ?  The  notion  of 
limits  may  give  an  answer ;  but  what  is  a  system  of  geometry  which 
introduces  continuity  and  limits  to  the  mind  as  yet  untaught  to  think 
of  space  and  of  magnitude  ?  " 

Benjamin  Peirce  says,  in  the  preface  to  his  geometry,  "  The  term 
direction  is  introduced  into  this  treatise  without  being  defined  j  but  it 
is  regarded  as  a  simple  idea,  and  to  be  as  incapable  of  definition  as 
length,  breadth^  and  thicJcnessJ^  But  in  case  of  length  we  have  clear  and 
rigorous  means  of  testiug  by  the  method  of  superposition  whether  two 
lengths  are  equal  or  unequal.    The  same  is  true  of  breadth  and  thick- 

*  Die  Nicht-Euclidisclien  Eatimformen  in  Analytischer  Behandlung,  Leipzig,  1835,  p.  iv. 
t  Athenaeum,  July  18,  1868. 


HISTOEICAL   ESSAYS.  383 

ness.  In  case  of  direction,  on  the  other  hand,  comparisons  cannot 
always  be  instituted,  at  least  not  without  becoming  involved  iu  logical 
difficulties.  We  have  no  satisfactory  means  of  telling  whether  two  non- 
intersecting  lines  in  a  plane  have  the  same  or  different  directions.  We 
are  not  even  sure  that  the  relation  of  direction  can  be  applied  to  them. 
On  a  pseudo-spherical  surface  a  whole  pencil  of  lines  can  be  drawn 
through  a  given  point  which  do  not  intersect  a  given  line.  The  lines  in 
this  pencil  do  not  have  the  same  directions  with  respect  to  one  another. 
The  question  then  arises,  which  one,  if  any,  of  these  lines  in  the  pencil 
has  the  same  direction  as  the  given  line  ?  If  we  can  not  distinguish 
between  the  presence  and  absence  of  a  quality,  then  that  quality  is 
useless. 

But  suppose  that,  for  the  sake  of  argument,  we  waive  the  above 
objection,  and  say  that  parallel  lines  have  the  same  direction.  After 
defining  straight  line,  angle,  parallel  lines,  in  accordance  with  the  concept 
of  "  direction,"  we  can  reason  in  the  same  way  as  Hayward  does  in  the 
quotation  given  under  the  third  heading.  But  that  mode  of  treating 
parallels  excludes  the  possibility  of  the  existence  of  pseudo-spherical 
geometry,  inasmuch  as  it  renders  absurd  the  statement  that  two  or 
more  lines  intersecting  one  another  may  exist,  none  of  which  intersect  a 
third  line,  for  lines  in  a  plane  which  have  different  directions  with  re- 
spect to  one  another  cannot  all  have  the  same  direction  with  respect  to 
a  third  line.  The  above  use  of  the  term  direction  involves  assumptions 
as  to  the  character  of  space  which  are  too  narrow  to  admit  the  use  of 
that  term  as  a  fundamental  concept.  As  far  as  possible,  our  Euclidiai] 
geometry  should  be  made  to  rest  upon  concepts  which  need  not  be 
abandoned  when  we  take  a  generalized  view  of  the  science.  Our  treat- 
ment of  the  elements  should  be  a  "  rock -foundation  for  more  advanced 
study." 

One  of  the  many  objections  to  all  attempts  to  found  the  elements  of 
geometry  on  the  word  "direction"  is  stated  by  Professor  Halsted  in  the 
following  manner:*  "Direction  is  a  common  English  word,  and  in 
Webster's  Dictionary,  our  standard,  the  only  definition  of  it  in  a  sense 
at  all  mathematical  is  the  fourth :  '  The  line  or  course  upon  which  any- 
thingis  moving  »  *  *  j  as,  the  ship  sailed  in  a  southeasterly  ^fra';f*o?i.' 
Direction,  to  be  understood  in  any  strict  sense  whatever,  posits  and 
presupposes  three  fundamental  geometric  ideas,  namely,  straight  line, 
angle,  parallels.  After  the  theory  of  parallels  founded  upon  an  explicit 
assumption  has  been  carefully  established,  a  strict  definition  of  direc- 
tion may  be  based  upon  these  three  more  simple  concepts,  and  we  may 
nse  it  as  Rowan  Hamilton  does  in  his  Quaternions.  But  in  American 
geometries,  for  example  Wentworth's,  the  fallacy  petitio  principii  is 
three  times  repeated  by  defining  the  three  component  parts  of  direction, 
each  by  direction  itself." 

Professor  Halsted  objects  also  to  the  word  "distance"  as  a  fuuda- 

*  Letter  to  the  writer,  November  17, 1688. 


384  TEACHING  AND   HISTORY   OP   MATHEMATICS. 

mental  idea.  He  says,  in  the  preface  to  his  elements,  that  the  attempt, 
"  to  take  away  by  definition  from  the  familiar  word  *  distance'  its  ab- 
stract character  and  connection  with  length-units,  only  confuses  the 
ordinary  student.  A  reference  to  the  article  '  Measurement,'  in  the 
Encyclopaedia  Britannica,  will  show  that  around  the  word  '  distance ' 
centers  the  most  abstruse  advance  in  pure  science  and  philosophy.  An 
elementary  geometry  has  no  need  of  the  words  '  direction '  and  '  dis- 
tance.' "  This  view  receives  support  from  Dr.  Killing  in  the  above  quo- 
tation. Professor  Halsted  has  introduced  the  new  word  sect,  meaning 
"  part  of  a  line  between  two  definite  points,"  and  corresponding  to  the 
German  word  Streeke.  The  objections  to  the  word  distance  are  stated 
by  him  in  the  following  words :  *  "Distance  is  also  a  common  English 
word,  and  Webster  as  its  first  definition  gives,  'An  interval  or  space 
between  two  objects ;  the  length  of  the  shortest  line  which  intervenes 
between  two  things  that  are  separate.  Every  particle  attracts  every 
other  with  a  force  *  •  *  inversely  proportioned  to  the  square  of  the 
distance.  Newton.'  Thus,  distance  posits  shortest  line  and  length,  there- 
fore measurement,  therefore  ratio,  never  treated  before  the  fifth  book  in 
the  Euclidian  geometry,  and  never  adequately  treated  at  all  in  any  other 
geometry  without  the  use  of  the  whole  theory  of  limits.  Yet  American 
geometries,  for  example  Wentworth's,  give  in  place  of  the  well-known 
simple  proof  of  the  theorem  that  any  two  sides  of  a  triangle  are  together 
greater  than  the  third,  the  abstruse  assumption  '  a  straight  line  is  the 
shortest  distance  between  any  two  points,'  and  that  after  having  ex- 
plicitly said  that  there  is  only  one  distance  between  two  points." 

Before  concluding  this  essay  we  should  like  to  express  our  belief  that 
detailed  discussions  of  the  fundamental  geometric  concepts  should  be 
avoided  with  students  beginning  geometry.  Such  discussions  can  be 
carried  on  with  more  profit  when  reviewing  the  subject  near  the  end  of 
the  course,  or  w^hen  beginning  the  study  of  non-Euclidian  geometry. 
In  this  connection  I  can  not  forbear  quoting  from  a  letter  of  Dr.  E.  W. 
Davis,  of  the  University  of  South  Carolina.  Says  he,  "  This  getting 
down  to  the  ultimate  basis  of  our  assumptions  is  a  long  and  painful 
process,  and  should  not  be  insisted  upon  in  elementary  instruction. 
The  first  beginning  in  mathematical  reasoning  should  be  reasoning  that 
shows  the  student  facts  that  are  new  to  him.  It  disgusts  him  to  have 
continually  i^ro-ye^  to  him  what  he  has  always  Icnown,  ov  to  hegin  by 
asking  him  to  doubt  what  he  can  not  help  but  deem  trtie  in  spite  of  all 
our  fine  logic.  Confidence  in  logic  should  be  gained  by  long  experience 
in  predicting  by  it  the  unforeseen,  before  we  proceed  by  it  to  invalidate 
deeply -rooted  and  universally  cherished  conceptions."  While  we  fully 
indorse  these  views,  we  at  the  same  time  insist  upon  a  scientific  treat- 
ment of  geometry  in  our  text-books,  for  the  two  following  reasons :  (1) 
When  the  student  advances  to  a  more  generalized  view  of  the  subject, 
he  will  find  that  his  first  studies  in  this  line  rested  upon  a  rock-founda- 
tion, and  that  the  old  edifice  can  bo  enlarged  without  being  first  de- 

*  Letter  to  the  -writer,  November  17,  1888. 


HISTORICAL   ESSAYS.  385 

molished ;  (2)  A  teacber,  like  an  honest  preacher,  prefers  to  teach  doc- 
trine which  is,  to  the  best  of  his  knowledge,  logically  and  philosophy 
ically  true. 

ON  TEE  FOUNDATION  OF  ALGFBBA. 

From  Peacock's  Eeport  to  the  British  Association,  in  1833,  on  the  Ee- 
cent  Progress  and  Present  State  of  Certain  Branches  of  Analysis  (p.  188) 
we  quote  the  following  words  :  "Algebra  was  denominated  in  the  time 
of  Newton  specious  or  universal  arithmetic,  and  the  view  of  its  principles 
which  gave  rise  to  its  synonym  has  more  or  less  prevailed  in  almost 
every  treatise  upon  this  subject  which  has  appeared  since  his  time.  In  a 
similar  manner,  algebra  has  been  said  to  be  a  science  which  arises  from 
that  generalization  of  the  processes  of  arithmetic  which  results  from  the 
use  of  symbolical  language ;  but  though  in  the  exposition  of  the  prin- 
ciples of  algebra  arithmetic  has  always  been  taken  for  its  foundation, 
and  the  names  of  the  fundamental  operations  in  one  science  have  been 
transferred  to  the  other  without  any  immediate  change  in  their  mean- 
ing, yet  it  has  generally  been  found  necessary  subsequently  to  enlarge 
this  very  narrow  basis  of  so  very  general  a  science,  though  the  reason 
of  the  necessity  of  doing  eo,  and  the  precise  point  at  which,  or  extent 
to  which,  it  was  done,  has  usually  been  passed  over  without  notice." 

From  the  same  Eeport  (p.  284=)  we  quote  the  following :  "  In  the 
early  part  of  the  last  century  the  algebra  of  Maclaurin  was  almost  ex- 
clusively used  in  the  public  education  of  this  country.  It  is  unduly 
compressed  in  many  of  its  most  essential  elementary  parts,  and  is  un- 
duly expanded  in  others  which  have  reference  to  his  own  discoveries. 

*  *  *  It  was,  subsequently,  in  a  great  measure  susperseded,  in  the 
English  universities  at  least,  by  the  large  work  of  Saunderson.  It  was 
swelled  out  to  a  very  unwieldy  size  by  a  vast  number  of  examples 
worked  out  at  great  length ;  and  it  labored  under  the  very  serious  defect 
of  teaching  almost  exclusively  arithmetical  algebra,  being  far  behind  the 
work  of  Maclaurin  in  the  exposition  of  general  views  of  the  science." 

There  was  indeed,  in  those  days,  some  opposition  at  Cambridge  (Eng- 
land) to  the  use  of  negative  quantities  in  algebra.  Among  Cambridge 
algebraists  who  argued  against  the  use  of  such  quantities  were  Baron 
Francis  Maseres  (fellow  of  Clare),  author  of  a  dissertation  on  the  nega- 
tive sign  in  algebra  (1758),  and  W.  Frend,  author  of  Principles  of  Alge- 
bra (1796-99).  Both  of  these  persons  set  themselves  against  Saunder- 
son, Maclaurin,  and  the  rest  of  the  world  j  for  they  rejected  negative 
quantities  no  less  than  imaginaries ;  and,  like  Eobert  Simson,  "  made 
war  of  extermination  on  all  that  distinguished  algebra  from  arithmetic."* 

The  algebras  studied  by  the  early  teachers  and  pupils  in  this  country 
were  all  English  works.    Maclaurin,  Saunderson,  Thomas  Simpson, 

*  ScholcB  Academicw :  Some  Account  of  the  Studies  at  English  Universities  in  the 
Eighteenth  Century,  by  C.  Wordsworth,  1877,  p.  68. 

881— No.  3 25 


386  TEACHING   AND   HISTOKY    OF   MATHEMATICS. 

Hutton,  Bonny  castle,  and  Bridge  were  authors  that  could  be  found  in 
the  libraries  of  our  American  x)rofessors  of  mathematics  As  i)ointed 
out  by  Peacock,  these  authors  began  their  treatises  with  arithmetical 
algebra,  but  gradually  and  disguisedly  introduced  negative  quantities. 

It  is  to  be  expected  that  our  early  compilers  of  algebra  and  writers 
on'  mathematics  should  imitate  the  English.  The  first  publication  in 
this  country  of  a  mathematical  work  which  cau,  perhaps,  lay  some  little 
claim  to  originality,  was  the  work  by  Jared  Mansfield,  entitled,  Essays, 
Mathematical  and  Physical,  Containing  j^ew  Theories  and  Illustrations 
of  some  very  Inportant  and  Difficult  Subjects  of  the  Sciences.* 

In  the  first  essay,  Mansfield  says  that  "  affirmative  quantities  are  to 
be  added,  negative  ones  to  be  subtracted."  Negative  quantities  "can 
never  exist  alone  or  independently  5  *  *  *  for  to  suppose  a  com- 
pound where  the  elements  have  been  all  exhausted  by  the  diminishing 
quantities,  and  something  still  left,  would  be  very  absurd.  This,  how- 
ever, may  be  the  case  apparently,  and  in  reality  no  absurdity  follow. 
Thus  the  case  above  mentioned,  8—12,  is  absurd  in  itself,  when  pure 
numbers  are  considered ;  but  an  algebraist  who  knows  how  often  the 
signs  are  changed  in  order  to  develop  the  unknown  quantity,  and  that 
the  quantities  are  often  assumed  without  knowing  on  which  side  the 
difference  lies,  views  this  expression  as  nothing  else  than  the  difference 
of  12  and  8,  or  as  12— 8  5  for  those  terms  which  have  the  sign  4-  prefixed 
to  them,  have  precisely  the  same  effect  on  those  to  which  the  sign  —  is 
prefixed,  as  those  which  have  the  sign— on  those  which  have  the  sign 
-f .  The  signs  are  totally  indifferent,  excepting  as  to  the  operations, 
and  where  no  operation  is  to  be  performed  they  are  to  be  neglected." 

These  views  suggest  an  algebra  purely  arithmetical,  which  finds  it  as 
impossible  to  give  a  clear  explanation  of  negative  quantities  as  it  would 
of  the  imaginary  V—l.  In  fact,  negative  quantities  are  the  true  "im- 
aginaries"  of  such  an  algebra. 

Day's  Algebra  contains  a  detailed  discussion  of  positive  and  nega- 
tive quantities.  "A  negative  quantity  is  one  which  is  required  to  be 
subtracted.  When  several  quantities  enter  into  a  calculation,  it  is  fre- 
quently necessary  that  some  of  them  should  be  added  together,  while 
others  are  subtracted.  The  former  are  called  affirmative  or  positive, 
and  are  marked  with  the  sign  +  5  the  latter  are  termed  negative,  and 
distinguished  by  the  sign  — ."  But  when  a  negative  quantity  is  greater 
than  a  positive,  how  can  the  former  be  subtracted  from  the  latter? 
"The  answer  to  this  is,  that  the  greater  may  be  supposed  first  to  exhaust 
the  less,  and  then  to  leave  a  remainder  equal  to  the  difference  between 
the  two."  The  interpretation  of  positive  and  negative  quantities  is  then 
given  by  employing  the  ideas  of  gain  and  loss,  ascent  and  descent,  north 
and  south  latitude,  etc. 

■*The  work  contains  eiglit  essays.  Their  titles  are  as  follows:  (1)  On  the  Use  of 
the  Negative  Sign  in  Algebra,  (2)  Goniometrical  Properties,  (3)  Nautical  Astronomy, 
(4)  Orbicular  Motion,  (5)  Investigation  of  the  Loci,  (6)  Flusionary  Analysis,  (7) 
Theory  of  Gunnery,  (8)  Theory  of  the  Moon. 


HISTORICAL  ESSAYS.  387 

The  treatment  of  this  subject  in  Day's  Algebra  iJj  essentially  the  same 
as  that  given  by  all  American  books,  excepting  those  of  recent  date. 
It  is  only  within  the  last  ten  or  fifteen  years  that  our  writers  on  alge- 
bra (such  as  Olney,  Wentworth,  Wells,  Thomson  and  Quimby,  Bow- 
ser, Newcomb,  Oliver,  Wait,  and  Jones,  Yan  Yelzer  and  Slichter), 
have  explicitly  assumed  the  existence  of  negative  as  well  as  positive 
quantities  at  the  very  beginning  of  their  text-books,  and  have  clearly 
explained  that  the  series  of  algebraic  numbers  is  assumed  as  going  out 
from  0  indefinitely  in  both  directions,  and  that  the  signs  +  and  —  are 
used  not  only  as  signs  of  operation  to  indicate  addition  and  subtraction, 
but  also  as  signs  of  quality  to  indicate  the  nature  of  the  quantities  as 
positive  or  negative. 

The  algebra  that  is  usually  found  in  our  school-books,  wherein  quan- 
tities are  considered  as  being  one  or  the  Other  of  two  diametrically 
opposite  kinds,  has  been  called  single  algebra.  It  differs  from  pure 
arithmetic  in  assuming  the  existence  not  only  of  positive,  but  also 
of  negative  quantities.  Neither  pure  arithmetic  nor  single  algebra  is 
perfect  in  itself,  since  each  leads  to  expressions  th  at  are  meaningless.  In 
pure  arithmetic,  a—b,  whenever  a<  &,  expresses  an  impossibility  and  is 
an  "imaginary  value."  In  single  algebra  this  ceases  to  be  impossible, 
but  there  we  are  led  to  another  impossibility,  another  "  imaginary," 

namely,  / — !• 

By  proceeding  one  step  further  in  our  generalization  we  come  to 
double  algebra,  in  which  the  existence  of  complex  quantities  (of  the  form 
a  ±  y'  —  i  b)  is  assumed.  Geometrically,  such  a  quantity  represents  a 
line  of  definite  length  in  some  one  definite  direction  in  a  plane.  This 
algebra  is  capable  of  giving  meaning  to  all  the  expressions  to  which  it 
leads  and  is,  therefore,  perfect  in  itself.  Some  of  our  recent  text-books 
(as  Wentworth's,  Bowser's,  Newcomb's,  Van  Yelzer  and  Slichter's,  and 
especially  Oliver,  Wait,  and  Jones's)  give  a  more  or  less  complete  ac- 
count of  this  kind  of  algebra  in  their  chapters  on  imaginaries.  Triple, 
quadruple  (quaternions),  and  other  multiple  algebras  have  been  in- 
vented. 

It  will  be  seen  that  the  true  foundations  of  algebra  have  not  been 
understood  before  the  present  century.  The  theory  of  imaginaries  in 
double  algebra  has  been  developed  chiefly  by  Argand,  Gauss,  and 
Cauchy.  The  philosophy  of  the  first  principles  of  algebra  has  been 
studied  by  Peacock,  De  Morgan,  Hankel,  and  others.  They  established 
the  three  great  laws  of  operations,  i.  e.,  the  distributive,  associative, 
and  commutative  laws.  A  flood  of  additional  light  on  this  subject  was 
thrown  by  the  epoch-making  researches  of  Hamilton,  Grassman,  our 
own  Peirce,  and  their  followers.  They  conceived  new  algebras,  whose 
laws  differ  from  the  laws  of  ordinary  algebra.* 

*  An  excellent  liistorical  sketch  of  Multiple  Algebra,  by  J.  W.  Gibbs,  of  Yale,  will 
be  found  in  the  Proceedings  of  the  American  Association  for  the  Advancement  of 
Science,  Vol.  XXXV,  1886, 


388  TEACHING  AND   HISTORY   OF   MATHEMATICS. 


DIFFERENCE  BETWEEN  NAPIER'S  AND  NATURAL  LOGA- 
RITHMS.'' 

The  term  "  Napierian  logarithms  "  lias  been  used  in  three  different 
senses :  (1)  as  meaning  N^apier's  logarithms,  or  the  ones  invented  by  him 
and  published  in  1614  in  his  Mirifici  Logarithmorum  Ganonis  Descrip- 
tio ;  (2)  as  a  synonym  for  "  natural  logarithms;"  (3)  as  conveying  the 
first  and  second  meanings  combined,  and,  thereby,  implying  that  the  nat- 
ural logarithms  are  the  ones  invented  by  IsTapier.  Though  this  last  use  of 
the  term  is  inadmissible,  because  the  logarithms  invented  and  published 
by  Napier  are  really  different  from  the  natural  logarithms,  it  has,  never- 
theless, been  the  most  prevalent ;  especially  has  it  been  prevalent  in 
this  country. 

An  examination  of  the  algebras  which  have  been  in  use  in  our  schools 
will  at  once  convince  us  that  this  error  has  been  very  general.  We  may 
consult  the  algebras  of  Eay,  Greenleaf,  Ficklin,  Schuyler,  Loomis,  Eob- 
inson,  F.  H.  Smith,  Hackley,  Davies,  Bowser,  Stoddard  and  Henkle, 
Thomson  and  Quimby,  and  many  others,  and  we  find  it  stated  either 
that  Lord  Napier  selected  for  the  base  of  his  system  e  =  2.718  .  .  .,  or 
that  he  assumed  the  modulus  equal  to  unity.  Either  of  these  two  state- 
ments is  equivalent  to  saying  that  the  logarithms  invented  by  Napier 
are  identical  with  the  natural  logarithms.  Some  authors  make  state- 
ments like  the  following  one,  taken  from  the  revised  edition  of  Wells's 
University  Algebra  (p.  363):  "The  system  of  logarithms,  which  has  e 
for  its  base,  is  called  the  Napierian  system,  from  Napier,  the  inventor 
of  logarithms." 

The  objection  to  statements  like  this  is  that  they  almost  invariably 
mislead  the  student.  What  inference  is  more  natural  than  that  Napie- 
rian logarithms  were  invented  by  Napier  1  Some  explanation  ought 
therefore  to  be  made  guarding  against  this  error. 

But  I  have  seen  only  two  American  books  doing  this,  namely,  J.  M. 
Peirce's  Mathematical  Tables,  and  Van  Velzer  and  Slichter's  Course  in 
Algebra  (of  which  a  preliminary  edition  has  just  appeared).  In  these 
two  books  the  truth  is  conveyed  in  plain  words  that  Napier's  logarithms 
differ  from  the  natural.  It  is  the  object  of  this  article  to  explain  that 
difference. 

It  is  important  to  note  that,  in  Napier's  time,  our  exponential  nota- 
tion in  algebra  had  not  yet  come  into  use.  To  be  sure,  Stifel  in  Ger- 
many and  Stevin  in  Belgium  had,  previous  to  this,  made  some  attempts 
at  denoting  powers  by  indices;  but  this  notation  was  not  immediately 
appreciated,  nor  was  it  generally  known  to  mathematicians,  not  even 
to  the  celebrated  Harriot,  whose  algebra  appeared  long  after  Napier's 
death.    It  is  one  of  the  greatest  curiosities  in  the  history  of  mathe- 


*  This  article  lias  been  published  in  the  Mathematical  Magazine,  Vol.  II,  No.  1,  and 
is  here  reproduced  -svith  some  very  slight  changes. 


I 


HISTORICAL   ESSAYS.  389 

matics  that  logarithms  should  have  been  coustructed  before  exponents 
were  used.  We  know  how  naturally  logarithms  flow  from  the  exponen- 
tial symbol,  but  to  Napier  this  symbol  was  entirely  unknown. 

The  interesting  inquiry  then  arises,  What  was  IJfapier's  treatment  of 
logarithms?    It  may  be  briefly  stated  as  follows: 

AC  B 

I 1 1 


D 


Let  J. B  be  a  line  of  definite  length,  DE  a  line  extending  from  D  in- 
definitely. Imagine  two  points  set  in  motion  at  the  same  time,  and 
witli  the  same  initial  velocity;  the  one  point  moving  from  D  toward  E 
with  uniform  velocity ;  the  other  from  Ato  B  with  a  velocity  decreasing 
in  such  a  way  that  when  it  arrives  at  any  position,  0,  its  velocity  is 
proportional  to  the  remaining  distance,  BG.  While  the  latter  point 
travels  a  distance,  AC,  suppose  the  former  to  move  over  the  space  DF. 
Napier  called  DF  the  logarithm  of  BC.  He  first  applied  this  idea  to 
the  calculation  of  a  table  of  logarithms  for  the  natural  sines  in  trigo- 
nometry. In  the  above  figure,  AB  would  represent  the  sine  of  90°  or 
the  radius,  which  was  taken  by  him  equal  to  10,000,000  or  10"^.  BG  would 
be  the  sine  of  an  arc,  and  DF  its  logarithm. 

'•'The  logarithm,  therefore,  of  any  sine  is  a  number  very  nearly  ex- 
pressing the  line  which  increased  equally  in  the  meantime,  while  the 
line  of  the  whole  sine  decreased  proportionally  into  that  sine,  both  mo- 
tions being  equal-timed,  and  the  beginning  equally  swift."* 

This  treatment  of  the  subject  is  certainly  very  unique.  Let  us  now 
establish  the  relation  between  these  Napierian  logarithms  and  our  nat- 
ural logarithms.    Let  m=AB,  x=DF,  y=BG,  then  AG=m—y.    The 

velocity  of  the  point  G  is     ^^^~^'=ry,  r  being  a  constant.     Integrat- 

ing,  we  have 

—Nat  log  y=rt+c. 

When  t=0,  then  y=77i,  and  c=— Nat  log  m.    The  velocity  of  the  point 
C  is  rm,  when  t=0.    Since  the  two  points  start  with  the  same  velcity, 

we  have  -—=rm  as  the  uniform  velocity  of  the  point  F.    Hence  x= 

(A/V 

rmt.    Substituting  for  t  and  c  their  values,  and  remembering  that,  by 
definition,  a7=Nap  log  y,  we  get 

Nap  log  y=m  Nat  log  - 

The  constant  m  was  taken  equal  to  10^    Substituting  we  get 

Nap  log  2/= 10^  Nat  log  — 

as  the  equation  expressing  the  relation  between  Napierian  and  natural 
logarithms.  crii;! 

•Definition  6,  p.  3,  of  Napier's  Mirifici  Logarithmorum  Canonis  Descriptio,  etc.,  1614. 


390  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

That  tJiere  is  a  difference  between  the  two  is  evident  at  once.    We 
easily  observe  the  characteristic  property  of  Napierian  logarithms,  that 
they  decrease  as  the  number  itself  increases.    This  property  alone  should 
have  been  a  sufficient  guard  against  declaring  the  two  systems  identi- 
cal.   The  ifapierian  logarithm  of  10'  is  equal  to  zero.    The  Napierian 
logarithms  of  numbers  smaller  than  it  are  positive;  those  of  numbers 
larger  than  it  are  negative,  or,  in  the  language  of  Najiier,  "  less  than 
nothing."    In  further  illustration  we  give  the  following: 
Nap.  log.    1  =  161  180  956.509 ;  Nat.  log.    1  =  0 
Nap.  log.    2  =  154  249  484.703 ;  Nat.  log.    2  =  0.6  931  472 
Nap.  log.  10  =  138  155  105.578;  Nat.  log.  10  =  2.3  025  851 

The  question  may  be  asked  what  &aseNapier  selected  for  his  system. 
We  answer  that  he  did  not  calculate  his  logarithms  to  a  base  at  all.  He 
never  thought  nor  ever  had  any  idea  whatever  of  a  dase  in  connection 
with  logarithms.  The  notion  of  a  base  suggested  itself  to  mathema- 
ticians later,  after  the  algorithm  of  powers  and  exponents,  both  inte- 
gral and  fractional,  bad  come  to  be  better  understood. 

If  we  inquire  what  the  base  to  the  logarithms  in  Napier's  tables 
would  have  been  had  he  used  one,  then  it  will  be  found  that  it  does  not 
coincide  with  the  natural  base  e,  but  is  very  nearly  equal  to  its  recip- 
rocal. In  theory,  that  base  is  exactly  equal  to  the  reciprocal  of  e,  as 
will  be  seen  from  the  following  relation,*  which  is  merely  another  form 

of  the  one  given  above, 

Nap  log  y 

=0)' 


J(_  _  /  1  \        10' 
10' 


The  base  -  would  not  lead  accurately  to  Napier's  logarithmic  figures, 

because  the  inventor's  method  of  calculation  was  necessarily  some- 
what rude  and  inexact.    The  modulus  of  his  logarithms  is  not  equal 

to  1,   but  nearly  equal  to  — 1.    If  the  base  were  exactly  -,  then  the 

modulus  would  be  exactly  -^  1 ;  for  the  modulus  of  any  system  of  loga- 
rithms is  the  logarithm,  in  that  system,  of  the  Napierian  base  e. 

The  first  calculation  of  logarithms  to  the  base  of  the  natural  system 
was  made  by  John  Speidell  in  his  New  Logarithms,  published  in  Lon- 
don, in  1616,  or  five  years  after  the  first  appearance  of  Napier's  loga- 
rithmic tables.t 

*  To  make  tlie  theory  of  exponents  applicable  to  Napier's  logaritbms,  it  becomes 
necessary  to  divide  the  number  y  by  10^,  otherwise  the  base  raised  to  the  zero  power 
would  not  be  equal  to  unity.  This  division  really  amounts  to  making  the  length  of 
the  line  AB  equal  to  1  instead  of  W.  If  this  be  done,  then  Najp.  log.  y  must  also  be 
divided  by  10',  so  as  to  retain  the  inventor's  conception  that  the  two  points  on  the 
lines  AB  and  BE,  respectively,  move  with  equal  initial  velocities. 

t  The  error  of  calling  the  Napierian  and  natural  logarithms  one  and  the  same  sys- 
tem has  been  wide-spread.  We  may  pardon  the  celebrated  Montucla,  the  eldest 
prominent  writer  on  the  history  of  mathematics,  for  making  this  mistake  (Montucla, 


HISTORICAL  ESSAYS.  391 


CIRCLE  SQUABEUS. 

It  would  be  strange  if  America  had  not  produced  her  crop  of  "  circle- 
squarers,"  just  as  otlier  countries  have  done.  Our  history  of  them  will* 
be  quite  incomplete.  We  have  not  gone  out  of  our  way  to  seek  the  ac- 
quaintance of  this  singular  race  of  "  mathematicians,"  nor  have  we 
avoided  them.  A  few  individuals  have  come  across  our  path,  and  we 
proceed  to  tell  about  them  for  the  benefit  and  edification  not  so  much  of 
mathematicians  as  of  psychologists.  The  mathematician  contemplates 
the  products  of  only  sound  intellects ;  the  psychologist  studies  also  the 
utterings  of  minds  that  are  or  seem  to  be  diseased. 

The  history  of  the  quadrature  of  the  circle  is  not  without  its  sober 
lessons  to  mathematicians.  It  extends  back  through  centuries  almost 
to  the  beginning  of  geometry  as  a  science. 

The  student  of  the  history  of  mathematics  is  impressed  by  the  fact 
that  this  science,  more  than  any  other,  has  always  been  a  progressive 
one.  He  does  not  find  a  period  in  authentic  history  during  which 
mathematics  was  not  cultivated  quite  successfully  by  some  nation  or 
other.  The  earliest  contributions  were  made  by  the  Babylonians  and 
Egyptians,  then  came  the  Greeks,  then  the  Hindoos,  then  the  Arabs, 
and  finally  the  Europeans.  Like  metaphysics,  mathematics  has  en- 
countered fundamental  problems  apparently  of  insurmountable  diffi- 
culty. But  it  has  generally  had  the  good  fortune  to  perceive  that  for- 
tifications can  be  taken  in  other  ways  than  by  direct  attack  with  open 
force  5  that,  when  repulsed  from  a  direct  assault,  it  is  well  to  reconnoitre 

Histoire  des  Maihematiques,  Tome  II,  Paris,  1758,  p.  21),  but  there  is  hardly  any 
excuse  for  a  modern  writer,  such  as  Hoefer  (^Histoire  des  Mathematiques  depuis  leurs 
Origines  jusqu^au  Commencement  dii  Dix-neuvieme  Siecle,  Paris,  1874,  p.  378),  for 
stumbling  over  the  same  stone.  The  diiFerence  between  the  two  systems  was  pointed 
out  in  Germany  by  Karsten  in  17C8,  Kapstner  in  1774,  and  Mollweide  in  1808,  but 
no  attention  was  paid  to  their  writings  on  this  subject.  A  lucid  proof  of  the  non- 
identity  of  the  two  systems  was  given  by  Wackerbarth  (^'  Logaritlimes  JETyperlioliques 
et  Logariihmes  Neperiens,'"  Les  Mondes,  Tome  XXVI,  p.  626).  The  French  mathema- 
tician Biot  wrote  likewise  on  this  subject  {Journal  des  Savants,  1835,  p.  259),  as  did 
also  De  Morgan  in  England  (English  Cyclopsedia,  Article  "Tables").  Still  more 
recently  attention  has  been  called  to  this  matter  by  J.  W.  L.  Glaisher  (Encyclo- 
psedia  Britannica,  9th  ed.,  Article  "Logarithms"),  and  by  Siegmund  Guenther  (  fTjiier- 
suclmngenzur  GescMclite  der  matliemaiischen  Wissenschaften,  Leipzig,  1876,  p.  271).  The 
writings  of  these  scientists  do  not  seem  to  have  received  the  attention  they  deserve, 
and  the  erroneous  notion  of  the  identity  of  Napierian  and  natural  logarithms  still 
continues  to  be  almost  universal. 

Napier's  original  works  on  logarithms  are  very  scarce.  The  Mirifici  LogaiHthmorum 
Canonis  Descriptio,  etc.,  Edinhurgi,  1614,  can  be  found  in  the  Congressional  Library 
in  Washington  and  in  the  Eidgway  Library  in  Philadelphia.  The  latter  library  has 
also  the  English  edition  of  the  above  work,  translated  by  Edward  "Wright  in  1616. 
"  So  rare  are  these  original  editions  that,  of  the  two  greatest  historians  of  logarithms, 
Delambre  never  saw  the  Latin  edition  and  Montucla  never  heard  of  the  English," 
(Mark  Napier's  Biography  of  Lord  Napier,  p.  379). 


392  TEACHING   AND   HISTORY   OP   MATHEMATICS. 

and  occupy  the  surrounding  country  and  discover  the  secret  paths  by 
which  the  apparently  unconquerable  position  can  be  taken.* 

From  this  we  can  draw  the  valuable  lesson  that  it  is  not  always  best 
to  "  take  the  bull  by  the  horns." 

The  value  of  this  precept  may  be  seen  by  giving  an  instance  in  which 
it  has  been  violated.  The  history  of  the  quadrature  of  the  circle  is  in 
point.  An  untold  amount  of  intellectual  energy  has  been  expended 
upon  this  problem,  yet  no  conquest  has  been  made  by  direct  assault. 
The  circle-squarers  existed  in  crowds  even  before  the  time  of  Ar- 
chimedes and  in  all  succeeding  ages  in  which  geometry  was  cultivated, 
down  even  to  our  own.  Affer  the  invention  of  the  differential  calculus 
abundant  means  were  introduced  to  complete  the  quadrature,  if  such  a 
thing  were  possible.  Persons  versed  in  mathematics  became  convinced 
that  the  problem  could  not  be  solved,  and  dropped  it.  But  those  who 
still  continued  to  make  attempts  upon  this  "  enchanted  castle,"  as  it 
was  supposed  to  be,  were  completely  ignorant  of  the  history  of  the  sub- 
ject, and  generally  misunderstood  the  conditions  of  the  problem.  "  Our 
problem,"  says  DeMorgan,t  "  is  to  square  the  circle  with  the  old  allow- 
ance of  means  :  Euclid's  postulates  and  nothing  more.  We  can  not  re- 
member an  instance  in  which  a  question  to  be  solved  by  a  definite 
method  was  tried  by  the  best  heads  and  answered  at  last  by  that  method, 
after  thousands  of  complete  failures." 

But  great  advance  has  been  made  on  this  problem  by  approaching 
it  from  a  different  direction  and  by  newly  discovered  paths.  Lambert, 
an  Alsacian  mathematician,  proved  in  1761  that  the  ratio  of  the  cir- 
cumference of  a  circle  to  its  diameter  is  incommensurable.  Only  nine 
years  ago  Lindemann,  a  German  mathematician,  demonstrated  that 
this  ratio  is  also  transcendental,  and  that  the  quadrature  of  the  circle 
by  means  of  the  ruler  and  compass  only,  or  by  means  of  any  algebraic 
curve,  is  impossible.^  He  has  thus  shown  by  actual  proof  that  which 
keen-minded  mathematicians  had  long  suspected,  namely,  that  the 
great  army  of  circle-squarers  have,  for  more  than  two  thousand  years, 
been  assaulting  a  fortification  which  is  as  impossible  to  be  torn  down 
as  the  firmament  of  heaven  is  by  the  hand  of  man. 

Now-adays,  a  person  claiming  to  have  solved  this  problem  is  ranked 
by  mathematicians  in  the  same  class  with  inventors  of  "perpetual  mo- 
tion," and  discoverers  of  the  "fountain  of  perpetual  youth."  A  very 
peculiar  characteristic  of  circle-squarers,  or  quadrators,  as  Montucla 
calls  them,  is  that  they  cannot  be  convinced  of  their  errors.  The  first 
American  quadrator  we  shall  mention  is  William  David  Clark  Murdock, 
who,  in  a  pamphlet  of  eight  pages,  bearing  no  date,  gives  a  Demonstra- 
tion of  the  Quadrature  of  the  Circle. 

The  next  man  on  our  list  is  John  A.  Parker,  whose  work  on  The  Quad- 

*  H.  Hankel,  Entwickeliing  der  Mathemaiik  in  den  letzten  Jahrlmndertm,  p.  16. 
t  English  Cyclopaedia ;  article,  "Quadrature  of  the  Circle." 
t  Mathematische  Annalen,  Band  XX,  p.  213. 


HISTOEICAL   ESSAYS.  393 

rature  of  the  Circle  (1851)  was  reviewed  in  the  New  Englander  of  Feb- 
ruary, 1852.  The  most  prominent  characteristics  of  this  work,  says  the 
reviewer,  are,  a  contempt  for  "algebra,"  and  a  grudge  against  "profes- 
sors." The  author  proves  that  all  geometers  from  Euclid  to  his  (Par- 
ker's) great  forerunner,  Seba  Smith,  have  been  but  blockheads  in  the 
very  A  B  C's  of  their  science.  He  solves  in  a  twinkling  the  vexed  prob- 
lem of  the  "three  bodies."  He  seems  ashamed  of  his  usher,  Seba  Smith, 
and  takes  him  to  task  for  "  stealiug  his  thunder."  Over  twenty  years  of 
experience  seem  to  have  made  him  no  wiser.  In  1874  he  republished 
his  book  of  over  three  hundred  pages  in  almost  exactly  its  original 
form. 

The  next  publication  is  the  following :  The  Regulated  Area  of  the 
Circle  and  the  A.rea  of  the  Surface  of  the  Sphere,  by  Charles  P.  Bou- 
che,  Citizen  of  the  United  States  of  North  America,  Cincinnati,  1854. 
It  covers  sixty  pages.  The  author  says :  "  Notwithstanding  the  per- 
fection at  which  mathematics  may  have  arrived  in  rectilineal  geometry, 
planimetry,  and  stereometry,  yet  with  regard  to  the  etirve  line,  as  well 
as  the  spherical  surface,  we  have  remained  in  great  darkness  till  it 
pleased  the  author  of  Spheres  to  afford  us  some  light  in  this  respect, 
and  from  a  source  little  expected,  i.  e.,  through  the  medium  of  a  plain, 
but  a  moderately  cultivated,  seeker  after  truth.  By  the  assiduous  ap- 
plication of  mind  and  the  blessing  of  God,  I  have  ultimately  succeeded 
in  correcting  some  great  errors  respecting  curvilineal  geometry.''^ 

The  next  circle- squarer  on  our  list  is  Lawrence  Sluter  Benson,  the 
author  of  a  geometry.  In  1879  he  published  in  New  York  a  work  called 
Mathematics  in  a  Dilemma,  in  which  he  also  gives  an  extremely  inter- 
esting history  of  his  efforts  on  this  subject.  He  says  that  after  com- 
pleting a  course  of  studies  at  college  in  1858,  and  while  residing  on  his 
former  place  in  South  Carolina,  his  mind  drifted  into  geometrical  ab- 
stractions. He  published  new  modes  of  demonstrating  the  quadra- 
ture in  1860  and  in  1862.  He  says  that  he  offered  "  one  thousand 
dollars  to  any  one  who  could  refute  the  result  which  I  gave  for  the  cir- 
cle, namely,  that  the  perimeter  of  its  equivalent  square  is  exactly  equi- 
distant heticeen  the  squares  circumscribed  and  inscribed  about  the  circle  ;  tjie 
sides  of  all  the  squares  being  respectively  'parallel.  This  ojBfer  and  demon- 
stration drew  me  in  many  discussions,  for  mathematicians  claim  them- 
selves able  to  prove  that  this  intermediate  square  is  just  equal  to  an 
inscribed  decagon  in  the  circle ;  whence  they  argue  that  I  make  the 
circle  too  small.  Committees  of  expert  mathematicians — professors  in 
colleges  were  selected  to  decide  this  issue ;  but  no  decision  was  made. 
Therefore,  in  1864,  while  the  Civil  War  was  raging,  I  ran  the  blockade 
and  visited  Europe,  and  laid  my  demonstration  before  scientific  socie- 
ties and  distinguished  mathematicians  there."  He  then  says  that  he 
returned  to  New  York  and  published  a  simplification  of  the  Elements 
of  Euclid,  with  the  repudiation  of  the  reductio  ad  absurdum.  He  says 
that  these  changes  met  the  approval  of  Professor  Docharty  of  the  Col- 


394  TEACHING   AND   HISTORY   OF   MATHEMATICS. 

lege  of  the  City  of  New  York,  and  that,  in  1873,  Charles  Davies  pub- 
lished a  book  where  he  also  repudiated  the  absurdum  reasoning.  "For 
nearly  twenty  years  mathematicians  and  myself  have  been  at  logger- 
heads on  the  issue  made  by  me  about  the  circle.  I  now  propose  to  set 
at  rest  all  doubts  against  the  demonstration  published  by  me  in  1860 
and  1862."  In  more  recent  years  Mr.  Benson's  efforts  to  revolutionize 
mathematics  have  been  unabated. 

Dr.  A.  Martin  tells  us  of  a  quadrator  who  deposited  with  him  a  man- 
uscript, in  1885,  proving  that  the  long  sought  for  ratio  is  exactly  3|f . 
Mr.  Faber,  the  writer  of  it,  distinguished  himself  also  in  other  branches 
of  mathematical  inquiry.  In  a  phamphlet  of  thirty-four  pages,  in  1872, 
'•'  Theodore  Faber,  a  citizen  of  the  United  States,'  New  York, "  makes 
the  "  extraordinary  and  most  significant  discovery  of  a  lacMng  link  in 
the  demonstration  of  the  world-renowned  Pythagorean  problem,  utterly 
disproving  its  absolute  truth,  although  demonstrated  as  such  for  twenty- 
three  centuries. "  In  justice  to  Mr.  Benson,  it  should  be  remarked 
that  he,  too,  is  waging  war  against  Euclid,  I,  47.  * 

*  Since  writing  tlie  above,  we  have  received  from  Dr.  Artemas  Martin  a  copy  of 
the  Notes  and  Queries,  Vol.  V,  Nos.  6  and  7,  June  and  July,  1888,  giving  a  Bibliogra- 
phy of  Cyclometry  and  Quadratures.  From  this  article  we  see  that  Theodore  Faber 
has  appeared  in  print  also  on  the  subject  of  the  quadature  of  the  circle.  The  article 
gives  over  twenty  publications,  besides  the  ones  mentioned  above,  by  American 
writers  who  believe  that  they  have  found  the  true  and  exact  ratio. 


APPENDIX. 


BIBLIOGEAPHY  OF  FLUXIONS  AND  THE  CALCULUS. 

TEXT-BOOKS  PRINTED  IN  THE  UNITED  STATES. 

HuTTON,  Charles.    Course  of  Mathematics,  in  two  volumes. 

American  editions,  revised  by  Robert  Adrain,  appeared  in  1812,  1816  (?),  and  1822. 
Evert  Duyckinck  brought  out  an  edition  in  New  York  in  1828.  Another  edition  ap- 
peared in  1831.  The  second  volume  contains  a  short  account  of  the  doctrine  of  flux- 
ions, using  the  Newtonian  notation. 

ViNCE,  Eev.  S.  The  Principles  of  Fluxions,  first  American  edition,  corrected  and  en- 
larged.   Philadelphia :  Kimber  &  Conrad.     1812,    Pp.  256. 

Employs  the  Newtonian  notation. 
Bezout.    First  Principles  of  the  Differential  and  Integral  Calc-lus,  or  the  Doctrine  of 
Fluxions,  from  the  Mathematics  of  Bezout,  and  translated  from  the  French  for  use 
of  students  of  the  University  at  Cambridge,  New  England.   Boston,  1824,  Pp.  195. 

This  book  forms  a  part  of  Farrar's  Cambridge  Mathematics.  It  is  the  first  work 
published  in  this  country  employing  the  notation  of  Leibnitz  and  the  infinitesimal 
method,  "  In  the  Introduction,  taken  from  Carnot's  B4jiexions  stir  la  Metaphysique  du 
Calcul  Infinitesimal,  a  few  examples  are  given  to  show  the  truth  of  the  infinitesimal 
method,  independently  of  its  technical  form."  This  is  done  by  explaining  that  there 
is  a  "  compensation  of  errors," 
Ryan,  James.     The  Differential  and  Integral  Calculus.    New  York,  1828.*    Pp.  328. 

"  The  works  which  I  have  principally  used  in  preparing  this  treatise  are  Lacrois, 
Lardner,  Boucharlat,  Gamier,  and  Du  Bourguet's  Difterential  and  Integral  Calculus ; 
Lagrange's  Caleul  des  Functions,  Simpson's  Fluxion's,  Peacock's  Examples  on  the 
Differential  and  Integral  Calculus,  and  Hirsch's  Integral  Tables"  (advertisement). 
The  first  section  of  the  book  is  given  to  "  preliminary  principles,"  in  which  the  three 
methods  of  Newton,  D'Alembert,  and  Lagrange  are  explained.  The  mgthod  adopted 
by  the  author  is  that  of  limits,  but  no  formal  definition  of  the  term  "limit "  is  given. 
The  symbol  (0),  indicating  the  absence  of  quantity,  is  everywhere  treated  with  the 
same  courtesy  and  implicit  confidence  as  though  it  were  actually  a  quantity.  The 
inquiry  as  to  whether  the  laws  of  analysis  are  really  applicable  to  this  foreign  in- 
truder into  the  society  of  actual  magnitudes,  or  whether  it  has  to  be  governed  by 
laws  of  its  own,  is  nowhere  deemed  necessary.  These  remarks  apply  with  equal 
force  to  other  works  on  calculus  and  to  works  on  algebra. 

Young,  J,  R,  Elements  of  the  Differential  Calculus,  comprehending  the  general  the- 
ory of  surfaces,  and  of  curves  of  double  curvature.  Revised  and  corrected  by 
Michael  O'Shannessy,    Philadelphia,  1833.    Pp.  255. 

In  the  preparation  of  this  American  edition,  the  editor  was  assisted  by  Professor 
Dod,  of  Princeton  College. 

In  the  explanation  of  the  process  of  differentiation,  he  makes  h  absolutely  zero,  in 
an  expression  like  this : 

yL:zy  =  Sx^^3xh-\-h"'. 

*  Eyan's  Calculus  is  now  a  rare  book.  The  copy  we  have  before  as  was  kindly  lent  to  us  by  Prof, 
W,  Eutherford,  of  the  University  of  Georgia, 

395 


396  TEACHING  AND   HISTORY   OF   MATHEMATICS. 

"In  both  tliese  cases  (of  wliicli  tliat  given  here  is  one),  as  indeed  in  every  otlier- 
tlie  respective  differential  co-efficients  are  only  so  many  particular  values  of  the  gen, 

eral  symbol  -,  to  wbicli  ^  ~^  always  reduces,  "wben  7i  =  0."    In  the  above  example 

_  =  3x*.  "The  expressions  -^  and  ~  have,  we  see,  the  advantage  over  the  symbol 
0  Ax  dy 

-,  of  particularizing  the  function  and  the  independent  variable  under  consideration 

and  this,  it  must  be  remembered,  is  all  that  distinguishes  ^  or  ^  fi-om— ,  for  dw, 

ax        dy  0 

d.z,  dx,  are  each  absolutely  0."  "These  differentials,  although  each  =  0,  have,  never- 
theless, as  we  have  already  seen  a  determinate  relation  to  each  other  (!) ;  thus,  in 
the  last  example,  this  relation  is  such  that  dy  =  26  (a  +  ix)  dx,  and,  although  this 
is  the  same  as  saying  that  0  =  26  (a  +  bx)  0 ;  yet,  as  we  can  always  immediately  obtain 

from  this  form  the  true  value  of  —  or  ^,  we  do  not  hesitate  occasionally  to  make 

0  ,      dx' 

use  of  it."  It  will  thus  be  seen  that  the  author  has  no  hesitation  whatever  in  break- 
ing up  the  differential  co-efificient. 

Young,  J.  E.  T]ie  Elements  of  the  Integral  Calculus,  with  its  applications  to  geome- 
try and  to  the  summation  of  infinite  series.    Revised  and  corrected  by  Michael 
O'Shannessy.    Philadelphia,  1833.    Pp.  292. 
Da  VIES,  Chakles.    Analytical  Geometry  and  Differential  and  Integral  Calculus.    18 — . 

Elements  of  the  Differential  and  Integral  Calculus.    1836. 

Several  editions  of  Davies'  calculus  appeared.    In  the  improved  edition  of  1843 

u' — u 
(pp.  17  and  18)  the  author  says  that  2ax  is  the  limit  toward  which  the  ratio  —j^ — 

=  2ax  -\-  ah  approaches  in  proportion  as  h  diminishes,  and  hence  "expresses  that  par- 
ticular ratio  which  is  independent  of  the  value  of  h."  Bledsoe  objects  to  this,  saying, 
"  Shall  they  (teachers)   continue  to  seek  and  find  what  no  rational  beings  have  ever 

found,  namely,  that  particular  value  of  — t—  which  does  not  depend  on  the  value  of 

hi  That  is  to  say,  that  particular  value  of  a  fraction  which  does  not  depend  on  its 
denominator!"  Davies  represents  by  dx  "the  last  value  of  h,  which  can  not  be 
diminished,  according  to  the  law  of  change  to  which  h  ot  xia  subjected,  without  be- 
coming 0."  "It  may  be  difficult,"  says  the  author,  "  to  understand  why  the  value 
which  h  assuBies  in  passing  to  the  limiting  ratio  is  represented  by  dx  in  the  first 
member  aud  made  equal  to  0  in  the  second."  To  this  Bledsoe  says  :  "  Truly  this  is  a 
most  difficult  point  to  understand,  and  needs  explanation.  For  if  fe  be  made  abso- 
lutely zero,  or  nothing  on  one  side  of  the  equation,  why  should  it  not  also  be  made 
zero  on  the  other  side  V  "  Why  should  '  a  trace  of  the  letter  x '  be  preserved  in  the 
first  member  of  the  equation  and  not  in  the  second  ?  The  reason  is,  just  because  dx  is 
needed  in  the  first  member  and  not  in  the  second  to  enable  the  operator  to  proceed 
with  his  work." 

As  regards  the  conception  of  the  term  "  limit,"  Davies  believed  that  a  variable 
actually  reached  its  limit.  "  The  limit  of  a  variable  quantity  is  a  quantity  toward 
which  it  may  be  made  to  approach  nearer  than  any  given  quantity,  and  which  it 
reaches  under  a  particular  supposition."* 

Davies  believed  that  by  the  definition  of  M.  Duhamel,  according  to  which  a  varia- 
ble never  reaches  its  limit,  there  seemed  to  be  placed  an  "  impassable  barrier"  be- 
tween a  variable  quantity  and  its  limit.  "If  these  two  quantities  are  thus  to  be 
forever  separated,"  says  he,  "  how  can  they  be  brought  under  the  dominion  of  a  com- 
mon law,  and  enter  together  in  the  same  equation  ?  "t 

*  Nature  and  Utility  of  Mathematics,  by  Charles  Davies,  Ne-w  York,  1873,  p.  291. 
t  Ibid.,  p.  326. 


BIBLIOGRAPHY   OF   FLUXIONS   AND   THE   CALCULUS.         397 

Pkirce,  Benjamin.    An  Elementary  Treatise  on  Curves,  Functions,  and  Forces.    Vol- 
ume I,  containing  analytic  geometry  and  the  differential  calculus.    Boston  and 
Cambridge,  1S41.    Pp.  301.    Volume  II,  containing  calculus  of  imaginary  quan- 
tities, residual  calculus,  and  integral  calculus.     Boston,  1846.    Pp.  290. 
The  method  followed  in  these  volumes  is  the  infinitesimal,  of  which  the  author  was 
a  great  admirer.    The  differential  co-efficienta  are  here  denoted  by  D,  D',  etc.    The 
second  volume  treats  of  many  rather  advanced  subjects,  such  as  imaginary  infinitesi- 
mals, imaginary  logarithms,  imaginary  angles,  the  imaginary  angle  whose  sine  ex- 
ceeds unity,  potential  functions,  residuals,  definite  integrals,  elliptic  integrals,  method 
of  variations,  linear  differential  equations,  Eiccati'a  equation,  and  particular  solutions 
of  differential  equations. 

Chukch,  Albert  E.  Flements  of  the  Differential  and  Integral  Calculus.  New  York, 
1842. 
This  is  in  many  respects  a  good  work,  but  the  explanation  of  fundamental  princi- 
ples therein  contained  is  too  brief,  and  fails  to  convey  a  philosophic  knowledge  of 
them.  The  difficulties  which  a  student  is  likely  to  encounter  in  a  treatise  like  this 
have  been  well  stated  by  a  writer  in  the  Nation  of  October  18,  1888  :  "  What  vexes 
and  perplexes  him  (the  student)  is  that  he  seems  to  himself  to  comprehend  very 
clearly  what  he  is  doing,  and  to  be  doing  what  all  his  previous  training  had  taught 
him  he  must  not  do.  It  all  seems  very  easy,  very  simple,  and  very  absurd.  He  is 
told  to  '  take  the  limit '  of  one  side  of  his  equation  by  striking  out  a  quantity  because 
it  '  is  approaching  zero,'  while  on  the  other  side  the  same  quantity  must  be  carefully 
preserved,  because  it  is  one  of  the  terms  of  the  ratio  which,  is  the  very  essence  of  the 
whole  process." 

McCartney,  "Washington.    Principles  of  the  Differential  and  Integral  Calculus,  and 

their  application  to  Geometry.    Philadelphia,  1844.    Pp.  340. 

The  author  makes  use  of  the  doctrine  of  limits,  but  retains  the  language  of  infini- 

dy  dy 

tesimals.     "  ^  is  i^sed  as  a  mere  symbol  to  denote  the  ultimate  ratio,  jr  being  in  reality 

jv.    But  inasmuch  as  the  rules  for  differentiating  and  the  geometrical  application  of 

ultimate  ratios  are  more  readily  understood  by  regarding  the  increments  of  the  ordi- 
nate and  abscissa  as  indefinitely  small,  we  will  call  these  increments  in  their  ultimate 
state,  indefinitely  small  quantities."  "  For  the  sake  of  convenience,"  the  student  is  asked 
to  call  dy  and  dx  what  he  has  just  been  told  that  they  really  are  not.  Such  an  expo- 
sition of  a  fundamental  principle  is  quite  apt  to  fail  to  give  satisfaction  to  beginners, 

McCartney's  Calculus  is  a  book  possessing  several  good  features. 

LooMis,  Elias.    Analytical  Geometry  and  Calculus.    1851.  \ 

Later  the  Calculus  was  published  in  a  separate  volume  and  much  enlarged.  The 
unfolding  of  fundamental  principles,  as  given  in  the  improved  edition  of  1874,  is  less 
objectionable  than  that  in  the  preceding  works  which  adopt  the  method  of  limits. 
The  term  "limit  of  a  variable  "  is  here  subjected  to  definition,  but  the  student  is  not 

informed  whether  or  not  the  variable  ever  reaches  its  limit.    The  symbol  j^  is  made 

incr,  y  ^ 

to  represent  the  limiting  value  of  jzr--—.    Confusion  is  apt  to  arise  in  the  mind  of  the 

student  from  the  fact  that  dxis  "  put  for  the  incr.  x  in  the  limiting  value  "  (which 
value  is  zero),  and  is  afterward  said  to  be  "indeterminate"  in  value,  "  either  finite 
or  indefinitely  small." 

Smyth,  William.    Elements  of  the  Diferential  and  Integral  Calculus.    1854. 

The  author  uses  the  infinitesimal  method,  but  says  (p.  229)  that  "  as  a  logical  basis 
of  the  calculus,  the  method  of  Newton  and  especially  that  of  Lagrange  have  some  ad- 
vantage. In  other  respects  the  superiority  is  immeasurably  on  the  side  of  the  method 
of  Leibnitz." 


398  TEACHING   AND   HISTOEY   OF   MATHEMATICS. 

COURTENAY,  EDWARD  H.     Treatise  on  the  Differential  and  Integral  Calculus  and  on 
the  Calculus  of  Variations.    New  York,  1855.    Pp.  501. 
The  exposition  of  the  method  of  limits,  as  given  in  this  in  many  respects  admirable 
work,  is  likewise  open  to  objection,    dx  is  pronounced  to  be  "indefinitely  small"  and 
equal  to  h,  but  when  ft  =  0  at  the  limit,  dx  continues  to  remain  indefinitely  email. 

EoBiNSON,  Horatio  N.    Differential  and  Integral  Calculus,  1861. 

Some  of  Eobinson's  elementary  works  on  mathematics  became  popular,  but  not  so 
his  advanced  works.  His  calculus  and  astronomy  met  with  able  but  severe  criticism 
in  the  Mathematical  Monthly.  Eobinson's  work  did  not  appear  in  a  second  edition, 
but  the  work  of  Quinby  was  added  to  "  Eobinson's  Series"  in  place  of  it. 

DOCHARTY,  Gerardus  Beekman.  Elements  of  Analytical  Geometry  and  of  the  Differ- 
ential and  Integral  Calculus.  New  York,  1865.    Pp.  306. 

The  part  on  the  calculus  covers  204  pages. 

The  method  of  limits  is  employed  and  treated  in  the  manner  customary  with  us  at 
thfe  time  the  book  was  written.  ^ 

Spare,  John.     The  Differential  Calculus  :  with  Unusual  and  Particular  Analysis  of  its 
Elementary  Principles,   and  Copious  Illustrations  of  its  Practical  Application. 
Boston,  1865.     Pp.  244. 
This  work  I  have  never  seen.    Dr.  Artemas  Martin,  who  kindly  sends  me  its  title 
calls  it  a  unique  work,  as  may  be  seen  from  the  following,  which  he  quotes  from  its 
preface:  "The  calculus  being  algebra,  a  strictly  numerical  science,  the  present 
treatise  claims  to  have  labored  successfully  in  putting  on  the  true  character  as  such. 
No  insinuation  is  allowed  to  prevail  that  it  is  any  part  whatever  of  analytical  geom- 
etry or  that  it  is  other  than  the  natural  sequel  and  supplement  of  common  algebra; 
useful,  indeed,  as  an  appliance,  to  borrow,  in  investigation  of  the  few  kinds  of 
geometrical  quantity." 

Quinby,  I.  F.    A  Xeiv  Treatise  on  the  Elements  of  the  Differential  and  Integral  Calculus. 
New  York,  1868.     Pp.  472. 
Here,  as  in  other  works  based  on  the  method  of  limits,  the  student  encounters  at 

the  outset  the  perplexing  statement  that  -k,  where  0  denotes  "  absolute  zero,"  is  equal 
to  some  particular  quantity. 

Strong,  Theodore.  A  Treatise  on  the  Differential  and  Integral  Calculus.  New  York, 
1869.  Pp.  617. 
This  work  was  printed,  but,  we  understand,  never  published.  The  author  died 
while  the  work  was  in  press.  Theodore  Strong  was  professor  at  Eutgers  College 
from  1827  to  1863,  and  enjoyed  the  reputation  of  being  one  of  the  very  deepest  and 
most  erudite  mathematicians  in  America.  He  was  a  very  frequent  contributor  to 
our  mathematical  periodicals.  To  students  who  possessed  taste  for  mathematical  in- 
vestigation he  was  a  good  teacher,  but  to  those  who  had  no  taste  he  was  unintelli- 
gible. He  had  an  unconscious  tendency  to  diverge  into  regions  where  the  ordinary 
student  could  not  follow  him.  This  same  tendency  is  exhibited  in  his  Calculus,  and 
also  in  his  Elementary  and  Higher  Algebra,  published  in  1859.  Both  works  possess 
many  original  features,  but  tlie  novelties  contained  in  them  are  not  always  improve- 
ments. These  books  are  defective  in  arrangmeeut,  and  not  at  all  suited  for  use  in 
the  class-room.  In  his  general  view  of  the  calculus  Strong  follows  Lagrange,  but  his 
mode  of  presentation  is  quite  new.  He  believed  that  his  treatment  divested  the  cal- 
culus of  all  its  old  metaphysical  encumbrances.  He  attempted  to  show  how  the 
foundations  of  this  science  could  be  established  without  the  intervention  of  any  of 
the  antiquated  hypotheses.  "  It  is  hence  cleai',"  says  he,  "  that  the  differential  and 
integral  calculus  are  deducible  from  what  has  been  done,  without  using  infinitesi- 
mals or  limiting  ratios  "  (p.  271). 


BIBLIOGRAPHY   OF   FLUXIONS   AND   THE   CALCULUS.         399 

Peck,  William  G.    Practical  Treatise  on  the  Differential  and  Integral  Calculus,  with  some 
of  its  applications  to  mechanics  and  astronomy.    New  York  and  Chicago,  1870. 
Pp.  208. 
Employs  the  infinitesimal  method. 

Sestini,  B.     Manual  of  Geometrical  and  Infinitesimal  Analysis.    Baltimore,  1871.    Pp. 
131.* 
The  infinitesimal  method  is  used. 

Olney,  Edward.    General  Geometry  and  Calculus.    New  York,  1871. 

The  part  on  the  infinitesimal  calculus  covers  152  pages.  The  infinitesimal  method 
is  used.  It  is  the  experience  of  the  large  majority  of  teachers  in  this  country  that 
the  infinitesimal  method,  taken  by  itself,  unaided  by  any  other  method,  does  not  seem 
rigorous  to  a  student  beginning  the  study  of  the  calculus,  and  does  not  fully  satisfy  Ms 
mind. 

Bice  and  Johnson.    The  Elements  of  the  Differential  Calculus,  founded  on  the  mecnod 
of  rates  or  fluxions.     (Printed  for  the  use  of  the  cadets  of  the  U.  S.  Naval  Acad- 
emy.)   New  York,  1874. 
Without  abandoning  the  ordinary  notation,  the  writers  return,  in  this  work,  to  the 
method  of  Newton.    Newton's  method  of  rates  or  fluxions  is  employed  in  subsequent 
treatises  written  by  the  same  authors,  and  also  in  the  works  of  Buckingham  and  Tay- 
lor.   By  these  writers  much-longed-for  improvements  in  the  philosophical  exposition 
of  the  fundamental  principles  of  the  transcendental  analysis  have  been  introduced. 

Johnson,  W.  Woolsey.    Integral  Calculus. 

Rice  and  Johnson.  An  Elementary  Treatise  on  the  Differential  Calculus,  founded  on 
the  method  of  rates  or  fluxions.    New  York,  1877.     Pp.  469. 

Rice  and  Johnson.    Differential  Calculus  (abridged). 

Rice  and  Johnson.    Differential  and  Integral  Calculus  (abridged). 

Clark,  James  G.    Elements  of  the  Infinitesimal  Ca^cmZms  (in  "Ray's  Series").    New 
York  and  Cincinnati,  1875.    Pp.  441. 
The  doctrine  of  limits  is  made  the  basis  of  this  work.    The  author  follows  mainly 
the  excellent  philosophical  treatise  of  M.  Duhamel. 

Buckingham,  C.  P.  Elements  of  the  Differential  und  Integral  Calculus.  By  a  new 
method,  founded  on  the  true  system  of  Sir  Isaac  Newton,  without  the  use  of  in- 
finitesimals or  limits.    Chicago,  1875. 

Byerly,  W.  E.    Elements  of  the  Differential  Calculus,  with  examples  and  applications. 
Boston,  1880. 
The  doctrine  of  limits  is  used  as  a  foundation  of  the  subject  and  preliminary  to  the 
adoption  of  the  more   convenient  infinitesimal  method.    The  notation  Dxy  is  em- 
ployed. 

Byerly,  W.  E.  Elements  of  the  Integral  Calculus,  with  a  key  to  the  solution  of  differ- 
ential equations.    BostOD,  1882. 

Bowser,  Edward  A.    An  Elementary  Treatise  on  the  Differential  and  Integral  Calculus. 
New  York,  1880. 
Adopts  infinitesimal  method. 

Taylor,  James  M.    Elements  of  the  Differential  and  Integral  Calculus.    Boston,  1884. 
The  author  employs  the  conception  of  rates. 


*  A  copy  of  this  work  was  lent  to  U8  by  Prof.  J.  F.  Dawson,  S.  J.,  of  Georgetown  College.  West 
WaBhington. 


400  TEACHING   AND   HISTOEY   OF   MATHEMATICS. 

Newcomb,  Simon.    Elements  of  the  Differential  and  Integral  Calculus.     New  York, 
1837. 
The  author  uses  the  method  of  infinitesimals,  hased  on  the  doctrine  of  limits.    An 
infinitesimal  quantity  is  here  defined  as  one  "in  the  act  of  approaching  zero  as  a 
limit."    This  definition  of  an  infinitesimal  has  now  been  very  generally  adopted. 

It  has  been  said  that  years  ago  a  cadet  at  West  Point,  extremely  fond  of  mathe- 
matics, thus  estimated  the  calculus:  "The  inventors  of  the  differential  and  integral 
calculus  have  claimed  that  this  branch  of  so-called  science  belongs  to  the  depart- 
ment of  mathematics,  and,  laboring  under  that  delusion,  have  introduced  it  into  the 
course  of  academical  instruction  for  the  torture  of  students.  Such  classification  is 
obviously  incorrect,  because  the  principles  of  mathematics  fall  within  the  scope  of 
the  reasoning  faculty.  The  calculus,  on  the  contrary,  lies  without  the  boundaries  of 
reason."*  That  such  should  have  been  the  impression  received  by  the  student  of 
the  early  works  on  calculus  is  not  at  all  strange.  Our  recent  publications  on  the 
subject  have,  however,  made  decided  progress  in  the  philosophical  exposition  of  the 
fundamental  principles.  With  a  modern  book  and  a  competent  teacher  there  is  no 
reason  why  the  ordinary  student  should  not  get  a  rational  understanding  of  the 
calculus. 

*Life  of  General  Natbauiel  Lyon,  p.  30.  The  passage  is  quoted  in  the  Analyst,  Vol. 1, 1874,  "Edu- 
cational  Testimony  Concerning  the  Calculas." 

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